Renegotiation-Proof Contracts with Persistent States*

Bruno Strulovici
May 31, 2022

Abstract
This paper introduces a new approach to model renegotiation in contractual relationships
and applies it to study how renegotiation shapes long-term contracts in principal-agent relationships with persistent states. The structure of players’ payoff and state dynamics generates analgebraic structure over contractual equilibria and determines the set of alternatives consideredthrough renegotiation. Using recent advances in functional stochastic differential equations, thepaper derives an Observability Theorem and a Revelation Principle to address asymmetric information over persistent variables in diffusion models. Truthful renegotiation-proof contracts arecharacterized by a single number—their sensitivity to the agent’s report—and are self-correcting
off the equilibrium path. The sensitivity of the optimal contract is increasing in information
persistence and decreasing in players’ patience.

1

Introduction

Financial bailouts, fiscal policies, and corporate pay cuts during recessions are instances in which
regulators, firms, and other economic agents have the possibility to renegotiate explicit or implicit
contracts in reaction to shocks in their environment. In these instances, moreover, state variables
such as agents’ revenue or productivity are often privately observed and correlated over time.
Despite its relevance for economics, renegotiation has proved challenging to model and analyze.
Even for repeated games, there is no universally accepted concept of “renegotiation-proof” equilibrium. One concept, internal consistency, is often perceived as a minimal requirement, and defined
as follows: an equilibrium is internally consistent if there do not exist two histories such that thecontinuation equilibrium following the first history is Pareto dominated by a continuation equilibrium following the second history. Presumably, players facing the first continuation equilibrium*

Northwestern University. This paper subsumes the 2011 working paper entitled “Renegotiation-Proof Contracts

with Moral Hazard and Persistent Private Information.” I am particularly grateful to Alex Bloedel, Hector Chade,Roozbeh Hosseini, R. Vijay Krishna, Nikolai Krylov, Kyohei Okumura, Alessandro Pavan, as well as to Bruno Biais,
Mehmet Ekmekci, Daniel Garrett, Brett Green, Christian Hellwig, Johannes Hörner, Francisco Ruiz-Aliseda, Larry
Samuelson, Ilya Segal, Itai Sher, Juuso Toikka, and various seminar audiences for questions and comments. Part ofthis research was conducted while I was visiting Kyoto University, who hospitality is gratefully acknowledged.

1

could think of moving to the second, Pareto superior continuation equilibrium, which would destroy
the initial equilibrium.1 Internal consistency is weak because it does not impose any comparison
with any other equilibrium. For example, indefinitely playing a bad Nash equilibrium is internallyconsistent.
At the opposite extreme, strongly renegotiation-proof equilibria (Farrell and Maskin (1989)) mustsustain comparisons with all internally consistent equilibria and may fail to exist. Other concepts
involve a fixed point problem determining the set of equilibria that are renegotiation proof (seeAsheim (1991)) and are subject to existence and multiplicity issues.
State Consistency: These problems would seem even more severe in environments, pervasivein economics, with persistent shocks. In these environments, however, a strengthening of internalconsistency can prove powerful enough to drastically reduce the set of renegotiation-proof equilibria
and yield sharp predictions about their properties. The idea is to compare continuation payoffs
of a given equilibrium not only across histories leading to the same state, but also across histories
leading to distinct states, thus creating a concept of state consistency.
For example, if a state variable describes the current productivity of a firm, one cannot directlycompare continuation equilibria starting from different productivity levels (or “states”) since theproduction frontier and, more generally, the physical environment is different across states. However, one may be able to transform a continuation equilibrium starting from a given state intoanother continuation equilibrium starting from a different state, for example by using some homotheticity argument. One may then compare the continuation payoffs of various transformations thatgenerate equilibria starting from the same state, and use Pareto dominance to determine whether an
equilibrium is vulnerable to renegotiation. State consistency is weaker than strongly renegotiationproofness because it does not require that players envision radically different equilibria, but onlythat they think by analogy with the equilibrium that they are playing.
State consistency imposes an algebraic structure on renegotiation-proof equilibria. By transforming
an equilibrium starting from one state into equilibria starting from other states, one generates anorbit within the set of equilibria, in a group-theoretic sense that the paper makes precise. Thisorbit has a structure that reflects the nature of the transformation. State consistency imposes a
comparison between continuations of an equilibrium and the orbit that it generates. As a result, astate-consistent equilibrium inherits many properties of its orbit. For example, in the simple modelof Section 2, the structure of the principal’s payoff inherits many properties from the structure of the
1

Internal consistency is due to Bernheim and Ray (1989) and closely related to Farrell and Maskin’s (1989) weakly

renegotiation-proof equilibria. Although internal consistency is a concept mild, Abreu and Pearce (1991) and Asheim(1991) point out that it does assume time invariance.

2

output dynamics and of the agent’s utility and effort cost functions, and the class of renegotiationproof contracts is reduced to a one-parameter family.
Private Information and Renegotiation: An additional complication arises if some persistentstate variables are observed only by some party. Even if parties can communicate, they may
misreport private information, and parties could have to renegotiate under asymmetric information.
One approach is to restrict attention to equilibria in which parties truthfully reveal their information. In a truthful equilibrium, all parties know the state variables at all times and state consistency
can be used to study truthful, renegotiation-proof equilibria.
With renegotiation, however, focusing on truthful equilibria may a priori be restrictive since theabsence of commitment precludes the use of the standard Revelation Principle. However, renegotiation also incites parties to try and erase any inefficiency in ongoing agreements, which requires
some exchange of information. When parties can communicate frequently, parties can renegotiateaway any inefficiency stemming from information asymmetries and cannot commit not to do so. If
efficiency requires full information disclosure, focusing on truthful equilibrium may thus be without
loss of generality when parties can continually communicate.2
Observability and Revelation Principle in a Diffusion Model: This paper takes a different
approach to address private information. It studies a continuous-time model, in which communication continually takes place. In the model, which is due to Williams (2011), an agent privately
observes a stochastic cash flow process, which he reports to a principal. The agent can lie: if
the true cash flow process is {Xt }t≥0 , the agent reports a cash flow process {Yt }t≥0 where Yt is afunction of {Xs }s≤t and of the agent’s reporting strategy L. Mathematically, the lying process L is
adapted3 to the filtration generated by X and is assumed to enter the reporting process Y throughits drift: dYt = dXt + Lt dt until Section 9, which considers reporting jumps.
The agent’s equilibrium strategy, L, is known to the principal. Therefore, given a reporting history
2

The main theorem of Strulovici (2017) formalizes this intuition: it shows in a principal-agent model with asymmet-

ric information that when (i) allocation efficiency requires full information disclosure and (ii) parties can communicate
with arbitrarily high frequency before the allocation takes place, information is fully revealed and the transaction isefficient. Maestri (2017) shows a similar result when parties transact in each period: information is revealed arbitrarily quickly relative to the parties’ discount rate as parties become arbitrarily patient. These papers consider an
explicit game of renegotiation—with offer and acceptance decisions—whereas renegotiation in the present paper ismodeled using a set-theoretic approach.
3
The requirement that the agent’s strategy is adapted to what the agent observes is standard in continuous-timemodels, including in Williams (2011). This assumption often plays a crucial role in the analysis, e.g., to apply theMartingale Representation Theorem (see, e.g., Sannikov (2008) and a large subsequent literature), but it is a priori
with loss of generality. See Section ??.

3

{Ys }s≤t the principal’s belief about Xt need not coincide with Yt . For example, if the agent often
underreports his cash flow in equilibrium, the principal’s belief about Xt will be higher than thereport Yt . The question, then, is to determine how much information the principal can back out
about X from observing the process Y and knowing the reporting strategy L. Since L can depend
arbitrarily on history, the question would seem a priori hard to answer.
Using recent advances in the analysis of functional stochastic differential equations,4 this paper
provides the following Observability Theorem: (i) Under very mild5 regularity conditions on L,
the principal can infer the true process X, (ii) Even without such conditions, there always exists a
belief equation based on the principal’s information, which, whenever it has a well-defined solution,
reveals the true cash flow X.
The Observability Theorem implies that the principal knows the agent’s cash flow and continuation
utility at all times and that parties have symmetric information at all times. This addresses thedifficulty raised by private information and validates the state consistency approach.
Building on this result, the paper provides the following Revelation Principle: any state-consistentequilibrium is outcome equivalent to a truthful state-consistent equilibrium, as long as the transformation group used to define state consistency satisfies a simple monotonicity condition.
Renegotiation-Proof Contracts with Persistent Private Information: The main application studied in this paper revisits a central question in macroeconomics:6 the agent who generates
the cash flow is risk averse and the principal is risk neutral. The agent reports and transfers his
cash flow to the principal and receives in exchange a transfer from the principal (equivalently, theagent is subsidized or taxed depending on his cash flow report). The principal proposes a contract(a transfer process adapted to the agent’s report process) to the agent, and an equilibrium consistsof a contract and a reporting strategy for the agent. The principal wishes to give the agent some
expected lifetime utility at the smallest possible cost. The basic tension is that, in order to properly
insure the agent, the principal must know the agent’s cash flow, but the agent may benefit fromunderreporting cash flow to get a higher subsidy (or lower tax) or, conversely, overreport his cash
flow to get rewarded by a higher continuation utility.
From the principal’s perspective there are two state variables: the current cash flow and the agent’s
continuation utility. When the agent has exponential utility, a natural class of equilibrium trans4
5

A functional SDE is an SDE whose drift and volatility at time t depend on the path of the solution until t.
The theorem provides two independent sufficient conditions: a local Lipschitz condition and an “arbitrarily small

delay” condition, which means that the agent’s lying strategy cannot depend on what he observed in the last ε unitsof time, where ε is arbitrarily small. This condition is particularly mild because the process X is continuous. The
theorem also requires a local boundedness condition.
6
See Thomas and Worrall (1990), Williams (2011), Bloedel, Krishna, and Leukhina (2020) and references therein.

4

formations from one state to another emerges, which creates a rich and well structured class ofchallengers for the equilibrium and its continuations.
Together with an additional comparison (to convex combinations of equilibria), state consistency
pins down the structure of all renegotiation-proof equilibria and reduces the family of renegotiationproof equilibria to a simple, low-dimensional family, which may be viewed as the quotient of theset of renegotiation-proof equilibria with respect to their orbits.
Any truthful renegotiation-proof contract (i.e., a contract with which truth-telling forms an equilibrium) is characterized by a single “sensitivity” parameter, which determines the agent’s incentive totruthfully report his cash flows. For such a contract, all contractual variables have exact formulas
as a function of the sensitivity parameter and the continuation utility of the agent. The sensitivity
parameter describes how the agent’s continuation utility varies with his reports, and can take any
value between 0 and the coefficient of absolute risk aversion of the agent.
The class of renegotiation-proof contracts contains the contract studied by Williams (2011) and isequivalent to the class of stationary contracts studied by Bloedel, Krishna, and Strulovici (2020),
who provide a self-insurance implementation of these contracts. The optimal renegotiation-proof
contract is obtained by maximizing a closed-form objective with respect to the sensitivity parameter,
which makes it easy to derive comparative statics of the optimal sensitivity parameter with respect
to information persistence, discounting, and risk. The agent’s flow utility optimally has a negative
drift for all parameters of the model, but the occurrence of immiserisation depends on the strengthof cash flow persistence and volatility (Bloedel, Krishna, and Strulovici).
Reporting Incentives and Reporting Jumps: Reporting incentives are linear for any arbitrary contract, which implies that the agent either is indifferent between telling the truth andlying, or wishes to lie at maximal (infinite) rate, either upwards or downwards. To account for
this, the agent’s strategy space is enlarged to allow jumps in the agent’s reports. For the contracts
characterized in this paper, there is a natural way to specify how such jumps affect the agent’s
continuation utility. The agent’s incentives are characterized by a Hamilton-Jacobi-Bellman equation with an impulse response component, which provides a new (in the contracting literature, tothe author’s knowledge) and simple way of dealing with the possibility of unbounded drift of thereporting process. This technique is used to derive in closed form the agent’s value function not
only on the equilibrium path, but also after any possible deviation.7
With truthful renegotiation-proof contracts, the agent wants to report cash flows truthfully not
7

Another approach, which does not use jumps is to allow the agent’s optimal strategy to be ill defined off theequilibrium path. In this case, the agent’s value function is only a viscosity supersolution of the agent’s HJB equation.
This approach is explained in Bloedel, Krishna, and Strulovici (2020).

5

only on the equilibrium path, but also after any possible deviation. If, there was any mistake inthe report, it is strictly optimal for the agent to immediately correct this mistake.8
Literature Review: This paper proposes a way to analyze renegotiation in contractual relationships with persistent state variables, and applies it to study environments with persistent private
information.9 The concept may be viewed as an adaptation and strengthening of the concepts ofinternal consistency introduced by Bernheim and Ray (1989) and Farrell and Maskin (1989) tosettings with persistent states.10 Gromb (1994) studies a binary-state model of debt contracts andcompares payoffs across the two states, similarly to the approach of this paper. Bassetto, Huo, andRios-Rull (2020) study “organizational equilibria” in a dynamic game played a succession of time
inconsistent agents, in which capital is a state variable. They impose a weak separability condition
on agents’ preferences, which stipulates that an agent’s preference over sequences of actions is independent of the current level of capital. This weak separability condition allows comparisons ofcontinuation equilibria across different levels of capitals. When comparing continuation equilibria,
they impose a “no-restarting” condition, which means that there is no time t such that agents’
would prefer to adopt an earlier continuation equilibrium over the current one. This condition may
be viewed as a one-sided notion of internal consistency.11 Ray (1994) proposes a reinforcement ofinternal consistency, which may be interpreted as follows: if parties are aware of all continuationsof an equilibrium (as they should under internal consistency), they could build new equilibria recursively by using the set of continuation payoffs to incentivize current-period actions (see also Van
Damme (1991)). Maestri (2017) studies explicit, dynamic renegotiation between a principal andan agent with a private type.12
The paper also builds on the literature on dynamic contracting with persistent private information
initiated by Fernandes and Phelan (2000). Most modeling features of this paper are based on
Williams (2011) who focuses on full commitment.13 The model is related to optimal insurance and8This feature is interesting, for instance, if the agent is a newly-arrived CEO who discovers, upon taking the job,that the financial situation of his firm is worse than what outsiders think. The contracts characterized here give theagent the incentive to correctly book a nonrecurring loss on the firm’s accounts.
9Hart and Tirole (1988), Laffont and Tirole (1990) Dewatripont (1989), Fudenberg and Tirole (1990), andBattaglini (2007) study contract renegotiation with private information in finite period models, which impose defacto constraints on the frequency of communication. Maestri (2017) studies an infinite horizon version of Hart andTirole (1988) and finds that information is revealed arbitrarily quickly relative to the discount rate as parties become
arbitrarily patient.
10
See also Pearce (1987), Abreu and Pearce (1991), and Abreu et al. (1993).
11
They also impose a no-delay condition, which has no equivalent in the present paper, and a notion of optimality
which is similar to external stability in Bernheim and Ray (1989).
12
Kranz and Ohlendorf (2013), Miller and Watson (2013), and Safronov and Strulovici (2018) consider explicit
renegotiation in repeated games.
13
See also Tchistyi (2006), Zhang (2009), and Kapička (2013). Doepke and Townsend (2006) and Fukushima and6taxation models studied by Green (1987), Thomas and Worrall (1990), Golosov et al. (2003), Farhi
and Werning (2013). Golosov and Iovino (2020) study optimal insurance without commitment.
Pavan, Segal, and Toikka (2014) study persistence in discrete time through “impulse response
functions.” Contracting with persistent private information also arises in delegated experimentation
models (Bergemann and Hege (2005), Garfagnini (2011), Hörner and Samuelson (2013)), as anagent’s private effort to learn about some technology may result in persistent superior information.14
The paper is organized as follows. Section 2 introduces illustrates the concept and implicationsof state consistency in a simple setting with discrete time and with a persistent but publicly
observable state. Section 3 presents the contractual setting with persistent private information.
Section 4 proposes a general formalization of state consistency. Section 6 characterizes truthful,renegotiation-proof contracts in the setting of Section 3. Section 7 studies how the sensitivity ofthe optimal contract to the agent’s reports varies with persistence, discounting, and risk. Section 8
establishes an Observability Theorem and a Revelation Principle to address private information.
Section 9 provides a necessary and sufficient condition for truthfulness on and off the equilibrium
path. Section 10 discusses several extensions.

2Discrete-Time Example with Persistent Public Information

2.1

Setting

An agent generates an output process {Xt }t∈N such that
Xt+1 = Xt + D(At , Zt )where {Zt }t∈N is a sequence of i.i.d. shocks, At ∈ A is the agent’s effort at time t, D is a real-valued
function, and X0 = x ∈ R is given.15 The agent incurs a cost of effort h(At ) at time t where h issome arbitrary function h : A → R. A contract C maps each history {X0 , . . . , Xt } to a consumption
level Ct ∈ R. The principal and the agent have the same discount factor ρ ∈ (0, 1). Given a contract
C, the agent chooses an effort process A mapping histories into effort levels. Formally, an effort
Waki (2009) provide novel numerical methods and analysis.
14
See also Guo (2016) and Halac, Kartik, and Liu (2016). Similarly in DeMarzo and Sannikov (2016), principal andagent both learn the agent’s skill but the agent privately observes his effort. Sannikov (2014) considers a moral hazard
problem in which the agent’s action have long-term consequences. Garrett and Pavan (2012, 2015) study managerial
compensation contracts when the type of the manager is persistent. These papers do not consider renegotiation.
15
The example can be generalized to Xt+1 = λXt + D(At , Zt ) where λ ∈ (0, 1]. In this case, the transformation
Φδ defined by equation (1) must be changed to Φδ (C)(X00 , . . . , Xt0 ) = C(X00 − δ, X10 − δλ . . . , Xt0 − δλt ), with a similarchange for the transformation of the agent’s effort strategy, and the term x/(1 − ρ) in the payoff formulas appearingin Propositions 2 and 3 must be replaced by x/(1 − ρλ). For expositional simplicity, we focus here on λ = 1.

7

process A is admissible if it is adapted to the filtration generated by the output process X. We let
A denote the set of admissible effort processes.
Given a contract C and an admissible strategy A, the agent’s expected discounted utility is"#XtV (C, A; x) = E
ρ (u(Ct ) − h(At ))|X0 = xt∈N

for some real-valued utility function u and the principal’s expected discounted payoff is"#XtΠ(C, A; x) = E
ρ (Xt − Ct )|X0 = xt∈NA pair (C, A) is a contractual equilibrium if A maximizes V (C, Ã; x) over all admissible effort
processes Ã ∈ A. A contractual equilibrium (C, A) starts from (x, w) if X0 = x and V (C, A; x) = w.

2.2

Internal Consistency

In the context repeated games, two well-known and essentially identical concepts of renegotiation
are internal consistency (Bernheim and Ray (1989)) and weakly renegotiation-proofness (Farrell and
Maskin (1989)). According to these concepts, an equilibrium is renegotiation-proof if it does nothave any continuation equilibrium that is Pareto dominated by another continuation equilibrium.
Although the concept internal consistency was originally introduced in the context of repeatedgames, it can be embedded into dynamic games with non-trivial state variables: say that an
equilibrium is internally consistent if there do not exist two histories leading to the same state suchthat the players’ continuation payoffs following the first history Pareto dominate those following
the second history.
In the context of Section 2.1, a contractual equilibrium (C, A) is internally consistent if there do notexist two histories {Xs : s ≤ t} and {Xs0 : s ≤ t0 } leading to the same output level Xt = Xt00 = x andthe same continuation utility w for the agent such that principal gets a strictly higher continuation
payoff after history {Xs0 : s ≤ t0 } than after {Xs : s ≤ t}.
Intuitively, if a contractual equilibrium violates internal consistency and the dominated history{Xs : s ≤ t} is realized, the principal would strictly benefit from switching to the equilibrium
continuation following {Xs0 : s ≤ t0 } and the agent would accept this switch, with a strict incentive to
do so if the principal shares even an arbitrarily small fraction of the surplus gained from the switch.
Internal consistency implies the following result, whose proof is straightforward and omitted. Givena contractual equilibrium (C, A) and history up to time t, let Wt denote the agent’s continuation
utility at time t.
8

Proposition 1 If (C, A) is internally consistent, there exists a function Π : R2 → R such that the
principal’s continuation payoff process satisfies Πt = Π(Xt , Wt ) for all t.

2.3

State Consistency

Beyond Internal Consistency: To see how internal consistency may be strengthened, it isuseful to revisit about its rationale. Internal consistency presumes that, after observing somehistory, the principal is able to recognize that she could use the continuation equilibrium following
another history to achieve a higher payoff. This cognitive ability should extend to other naturalcomparisons.
One such comparison, often used in economics, is convexification: if two distinct continuation
equilibria give the same expected utility to the agent, and the agent has a concave utility, it isnatural for the principal to consider convex combinations of these continuation equilibria, as they
may achieve the same expected utility for the agent at a lower cost for the principal. This idea will
be exploited in Section 6.2 to narrow down the set of renegotiation-proof equilibria. For now, we
focus on comparisons across states, which constitute the main conceptual innovation of the paper.
Transformations Across Outputs: Fix a contractual equilibrium (C, A) starting from (x, w)
and another output level x0 ∈ R, and let δ = x0 − x ∈ R.
We define a new contract Φδ (C) byΦδ (C)(X00 , . . . , Xt0 ) = C(X00 − δ, . . . , Xt0 − δ).

(1)

We also define a new effort strategy Φδ (A) byΦδ (A)(X00 , . . . , Xt0 ) = A(X00 − δ, . . . , Xt0 − δ)(2)for any history {X00 , . . . Xt0 }.
The pair (Φδ (C), Φδ (A)) forms a contractual equilibrium starting from (x0 , w), which may be see asfollows. Consider any strategy A0 starting from x0 . Given the contract Φδ (C) and initial condition
x0 , A0 generates a stochastic output path X00 , X10 , . . . with X00 = x0 and results in a consumption
path given by (1).
Consider now the effort strategy Φ−δ (A0 ) obtained by applying transformation (2) to strategy A0
and shift parameter −δ rather than δ: for any history (X0 , . . . Xt ), we have Φ−δ (A0 )t = A0 (X0 +δ, . . . , Xt + δ). Notice that for any strategy A and δ ∈ R, we have Φ−δ (Φδ (A)) = A.

9

For any realization of the exogenous uncertainty (Z0 , Z1 , . . .), a simple forward induction on time t,
starting from t = 0, shows the following: the strategies A0 and Φ−δ (A0 ), starting respectively fromx0 and x are such that A0t = Φ−δ (A0 )t and Xt0 = Xt + δ for all t.
From (1), this implies that the agent receives the same consumption for all t in both of these cases.
Since the agent’s effort and consumption is the same across both contracts for each realization of
the shocks Z = (Z0 , . . .), we conclude that
V (Φδ (C), A0 |x + δ) = V (C, Φ−δ (A0 ); x)
for all A0 . Equivalently,
V (Φδ (C), Φδ (A)|x + δ) = Vx (C, A|x)
for all A. Therefore, we conclude that A is optimal given C and x if and only if Φδ (A) is optimal
given Φ(C) and x0 .
Continuation of a Contractual Equilibrium: In order to compare continuations of contractual
equilibria, it is useful to treat formally each such continuation as a contractual equilibrium withtime reset to 0.
Formally, given any contractual equilibrium (C, A) and any history {X0 , . . . , Xt } leading to somecontinuation utility Wt = w and output Xt = x, the continuation (Ĉ, Â) of (C, A) following history{X0 , . . . , Xt } is a contractual equilibrium starting from X̂0 = x and Ŵ0 = w, and such that for anyτ ≥ 0 and (X̂0 , . . . .X̂τ ), we haveĈ(X̂0 , . . . , X̂τ ) = C(X0 , . . . , Xt−1 , Xt = X̂0 , X̂1 , X̂2 , . . . X̂τ )andÂ(X̂0 , . . . , X̂τ ) = A(X0 , . . . , Xt−1 , Xt = X̂0 , X̂1 , X̂2 , . . . X̂τ ),where we indicated for clarity that Xt = X̂0 to emphasize that the end of history {X0 , . . . , Xt } isthe beginning of (Ĉ, Â)’s history.
Consistency Across Outputs:
A contractual equilibrium (C, A) is consistent across outputs if the following holds for any two
histories {X0 , . . . , Xt } and {X00 , . . . , Xt00 } leading to the same continuation promised utility Wt for
the agent. Denoting by Xt = x and Xt00 = x+δ the output levels after these respective histories, andletting (Ĉ, Â) denote continuation of (C, A) after {X0 , . . . , Xt }, the principal’s continuation payoff
after {X00 , . . . , Xt00 } is weakly greater than her payoff under the transformation (Φδ (Ĉ), Φδ (Â)) of
the continuation contract (Ĉ, Â).
Intuitively, consistency across outputs means that the principal can think, after history {X00 , . . . , Xt00 },of using the transformation of the continuation following {X0 , . . . , Xt } that makes this continuation
10

compatible with output level x + δ, and switch to this transformation if it is profitable. Consistency
across outputs implies internal consistency, as can be seen by setting δ = 0 in the definition.
Proposition 2 If (C, A) is consistent across outputs, then (i) the principal’s payoff function Π isadditively separable: there exist functions f and g such that Π(x, w) = f (x) − g(w) and (ii) f isxlinear in x: f (x) = 1−ρ
.

Proof.

Fix w ∈ R and consider any x 6= x0 such that the states (x, w) and (x0 , w) can both bereached by some histories. Given a continuation (C, A) starting from (x, w), the transformationof (C, A) constructed above for δ = x0 − x starts from (x0 , w) and gives the principal an expected0−x, because the expected output level at all future periods is uniformly
payoff equal to Π(x, w) + x1−ρ0−x.
translated by x0 − x. Consistency across outputs then implies that Π(x0 , w) ≥ Π(x, w) + x1−ρ0−x
The reverse transformation implies the reverse inequality. Hence, Π(x0 , w) = Π(x, w) + x1−ρforall (x, w) and (x0 , w) reachable by some histories. This implies that there exists some real numberx− g(w) for all x such that (x, w) can be reached. For x00 such thatg(w) such that Π(x, w) = 1−ρ

(x00 , w) is not reached by any history, we can define the function Π(x00 , w) using the same formula.
Since this transformation works for each w, we get the desired result.
Transformation Across Promised Utilities: We now specialize the setting further, to createtransformations across promised utilities of the agent. We now assume that consumption and effort
levels are nonnegative at all times, that the agent’s utility and cost functions are given by u(c) = cγ1
and h(a) = aγ2 for some parameters 0 < γ1 < 1 < γ2 , and that the output shocks take the following
multiplicative form:16
Xt+1 = Xt + At Zt .
Under these assumptions, the agent’s utility in any continuation of any contractual equilibrium
must be nonnegative, because the agent can always abstain from putting any effort and receive atleast zero utility from consumption in each period.
Given any contractual equilibrium (C, A) starting from state (x, w) and any β > 0, we now constructa contractual equilibrium (C 0 , A0 ) starting from (x, βw).
We start with the following observation: since the initial condition x is fixed and known to theprincipal, observing the output process (X0 , X1 , . . .) is equivalent to observing the increments (D0 =A0 Z0 , D1 = A1 Z1 , . . .). We can therefore express C and A as functions
Ct (D0 , . . . , Dt−1 )16

The example extends easily to the dynamics Xt+1 = Xt + dAt Zt where d is a positive constant. See alsofootnote 15.

11

and
At (D0 , . . . , Dt−1 )for all t.
0 ,We define a new contract starting from (x, βw) as follows. Given observed increments D00 , D10 , . . . Dt−1

let00
φβ (C)t (D00 , D10 , . . . , Dt−1
) = β 1/γ1 Ct (D00 β −1/γ2 , . . . , Dt−1
β −1/γ2 ).

We also define a new strategy φβ (A) for the agent, by00φβ (A)t (D00 , D10 , . . . , Dt−1
) = β 1/γ2 At (D00 β −1/γ2 , . . . , Dt−1
β −1/γ2 ).

We now show that (φβ (C), φβ (A)) is a contractual equilibrium starting from (x, βw) if and only if(C, A) is a contractual equilibrium starting from (x, w).
Consider any strategy A0 starting from initial condition (x, βw) and any realization of the exogenous
uncertainty (Z0 , Z1 , . . .), the strategy A0 defines an increment process D0 = (D00 = A00 Z0 , D10 =A01 Z1 , . . .). Consider the strategy φ1/β (A0 ) defined by
φ1/β (A0 )t (D0 , . . . Dt−1 ) = β −1/γ2 A0t (D0 β 1/γ2 , . . . , Dt−1 β 1/γ2 )(3)for all t. Notice that for all A and β > 0, we have φ1/β (φβ (A)) = A.17
By construction, the strategies A0 and φ1/β (A0 ) generate increment processes D0 and D such that
Dt0 = β 1/γ2 Dt .
for all t ≥ 0. Moreover, by definition of the contract φβ (C), the agent’s consumptions at time t
under (C, φ1/β )(A)) and (φβ (C), A0 ) are related by
φβ (C)t = Ct β 1/γ1

(4)

for all t. This implies that the agent’s utility from consumption and cost of effort satisfy u(φβ (C)t ) =βu(Ct ) by (4) and h(A0t ) = βh(φ1/β (A0 )t ) by (3) at all times. Since these equalities hold for all
realizations of the exogenous shocks, we conclude thatV (φβ (C), A0 ; x) = βV (C, φ1/β (A0 ); x)
for all A0 . Equivalently,
V (φβ (C), φβ (A); x) = βV (C, A); x).
17

Formally, this means that the map φ : (β, A) 7→ φβ (A) defines an action of the group ((0, +∞), ×) over the set

A of admissible strategies. This point is developed in Section 4.

12

for all A. This shows that (C, A) is a contractual equilibrium starting from (x, w) if and only if(φβ (C), φβ (A)) is contractual equilibrium starting from (x, βw).
State Consistency: We define state consistency in two steps. First, say that a contractual
equilibrium (C, A) is consistent across promised utilities if the following holds for any two histories{X0 , . . . , Xt } and {X00 , . . . , Xt00 } leading to the same output level Xt = Xt00 = x: denoting by
w and βw the continuation utility levels right after these histories, the principal’s continuation
payoff after {X00 , . . . , Xt00 } is weakly greater than her continuation payoff under the transformation(φβ (Ĉ), φβ (Â)) of the continuation (Ĉ, Â) of (C, A) after {X0 , . . . , Xt }.
Second, say that a contractual equilibrium is state consistent if it is consistent across outputs andacross promised utilities.
Proposition 3 Suppose that the contractual equilibrium (C, A) is state consistent. Then, there
exist positive constants α1 , α2 such that the principal’s payoff function Π satisfies
Π(x, w) =x+ α2 w1/γ2 − α1 w1/γ1 .
1−ρThe next proposition shows that state-consistent contractual equilibria and can be constructed
explicitly.
Proposition 4 There exists a continuum of state-consistent equilibria, which have the following
form:18 letting Wt denote the agent’s continuation utility at time t, we have
1/γ2

At = a1/γ2 Wt

1/γ1

Ct = c1/γ1 Wt
where a and c are positive constants related by(γ2 − 1)a = 1 − c.

(5)

In these equilibria, the effort and consumption levels at any time are only a function of the agent’s
continuation utility at that time, and these functions are pinned down by a one dimensional parameter (c, say), which fully determines, together with (5), each contract. To fully characterize thecontract, one has to determine how the promised utility Wt depends on observed output. This isdone in the proof of Proposition 4 and this dependence is also pinned down by c.
18

The statement concerns a specific class of state consistent equilibria. It does not rule out the existence of otherstate consistent equilibria.

13

3Persistent Private Information

We now turn to the main analysis of this paper, which includes persistent private information andformalizes the construction of transformation groups introduced in the example.
An agent generates cash flow Xt ∈ R at time t ≥ 0, which evolves according to the dynamic
equation19
dXt = [(ξ − λXt )] dt + σdBt

(6)where B is the standard Brownian motion and X0 = x is given and commonly known. The cash
flow has a mean-reversion component with speed λ and long run average ξ/λ. A low (high) meanreversion speed λ results in high (low) persistence of the cash flows and, hence, of the agent’s private
information. λ is the rate at which a shock in the current cash flow decays over time. Uncertainty
is modeled with a probability space (Ω, F, P ) satisfying the usual conditions and whose outcomes ω
are identified with the paths of B.
The agent reports and transfers to the principal a cash flow Yt that obeys the dynamic equation
dYt = dXt + Lt dt = [(ξ − λXt ) + Lt ] dt + σdBt

(7)

where Lt is the rate at which the agent lies about the increment dXt of the true cash flow.20
The gap Gt = Yt − Xt between reported and actual cash flows satisfies21
Z t
Gt =Ls ds.
0

The agent’s initial cash flow is assumed to be known by the principal.
Assumption 1 (i) The principal observes the report process {Yt }t>0 but not the actual cash flow
process {Xt }t>0 . (ii) Y0 = X0 .
Contract: A contract is a stochastic process C = {Ct }t≥0 that is adapted to the filtration FY ={F Y }t≥0 generated by the report process Y . The process C, discounted at rate r, is assumed to beR∞
integrable when the agent tells the truth: E[ 0 e−rt |Ct |dt |L ≡ 0] < ∞.
19

The model is due to Williams (2011). An earlier version of this paper (Strulovici (2011)) includes moral hazard:

the agent privately chooses some effort level At ∈ R that affects the cash flow dynamics according to dXt = (At +ξ − λXt )dt + σdBt . This addition does not affect the main results, as explained in Section 10.2.
20
The rate at which the agent lies may be unbounded. To address this, Section 9 allows jumps in the agent’s report
process and proposes a natural extension of the contract to this case.
21
One could impose constraints on the lies that the agent can make, e.g., by requiring that Yt ≤ Xt (the agent
cannot transfer more than he earns) or that Gt ≤ ḡ for some arbitrary ḡ > 0 (the agent cannot overreport more thana fixed amount). The contracts derived in this paper remain truthful in the presence of such constraints and for anyḡ > 0, no matter how small, the necessary conditions for truth-telling are the same as if G is unconstrained.

14

A contract C requires the principal to make a transfer Ct ∈ R to the agent at time t: the agent
gives Yt to the principal and the principal gives Ct to the agent. An alternative interpretation isthat the agent reports and keeps Yt and the principal gives a (possibly negative) subsidy Ct − Yt
to the agent. These interpretations are formally equivalent and the former is used throughout the
paper for consistency.
We rule out private savings and assume that the agent immediately consumes the transfer that hereceives from the principal, as well as any difference between his real and reported cash flows:22
Assumption 2 The agent’s consumption at time t is equal to Ct + (Xt − Yt ) = Ct − Gt .
The agent’s strategy consists of a lying process L adapted to the agent’s information, which corresponds to the filtration FX = {FtX }t≥0 generated by X.23
Given an initial condition x, a contract C, and a strategy L, the agent’s expected discounted utility
is24

Z ∞
V0 (C, L; x) = E

−rt

e


(u(Ct + Xt − Yt )) dt(8)0

where u is strictly concave and X, Y evolve according to (6) and (7) and the initial conditions
X0 = Y0 = x. In computations to follow, the agent will be assumed to have exponential utilityu(c) = − exp(−θc) for some risk-aversion coefficient θ > 0.
A strategy L is admissible if the following transversality or “No-Ponzi” condition is satisfied. Let
VT (L) denote the agent’s continuation utility at time T when control L is applied until time T andtruthtelling is used thereafter.25
lim E e−rT |VT (L)| = 0 a.s.

T →+∞

(9)

Definition 1 Given an initial condition x, a contractual equilibrium (C, L) consists of a contract
C and a strategy L for the agent that solves
sup V0 (C, L0 ; x)L0over all admissible strategies.
22
23

See Bloedel, Krishna, and Strulovici (2020) for the case of hidden savings.
The agent also observes Y . However, since X determines Y , given the agent’s strategy, X is a sufficient statistic

for the agent’s information.
24
The strategy L affects the probability measure over the paths of Y , which affects the expectation.
25
Admissibility rules out, for instance, strategies in which the agent continually underreports his cash flow, which
permits him to get a higher immediate transfer from the principal but leads to an ever-decreasing continuation utility,with a decrease rate that exceeds r. Condition (9) is standard. It is used to take limits in verification arguments, tocheck that the solution of an HJB equation is equal to the value function of the agent’s optimization problem.

15

A contractual equilibrium requires that the strategy L be incentive compatible given the contract
C and initial condition x. The contract C is not required to satisfy any incentive compatibility
condition for the principal.
Objective of the Principal: Given an initial condition x and a contractual equilibrium (C, L),
the principal’s expected payoff is
Z ∞
E

−rt

e


(Yt − Ct )dt0

where Y is given by (7), X follows the dynamic equation (6), and X0 = Y0 = x. The objective of
the principal is to maximize her expected payoff subject to giving the agent some minimal expected
lifetime utility w (i.e., V0 ≥ w) and to renegotiation-proofness constraints that are the focus of thenext section.

4

Concept of Renegotiation

Any model of renegotiation entails a comparison between some current agreement and alternative
agreements. The key is to determine the set of challengers that parties may consider as valid
alternatives to the current agreement. This section proposes an approach to do this when there are
persistent state variables.

4.1

Cash Flow and Continuation Utility as State Variables

In our setting, one of the state variables is the current cash flow Xt generated by the agent. Since
this cash flow is persistent, it affects the set of achievable payoffs for the players at any given time.
In addition, it is useful to treat the agent’s continuation utility Vt as a state. We take the perspectiveof the principal and consider whether there exist contractual equilibria that give the agent the same
utility Vt as the current equilibrium but increase the principal’s payoff Πt given the current cash
flow Xt .

4.2

Transformation Groups

We formalize state-consistency using the language of group theory. This formalization makes it
easy to envision the application of the concept of state consistency beyond the specific applications
considered in the present paper.
Let
16 S denote the state space. In our application, S consists of all pairs (v, x) of continuation

utility and cash flow over the relevant domain.
 G denote a group—in the algebraic sense—that exerts a left action on S. This means thatfor any g ∈ G and any (v, x) ∈ S, one can associate an element (ṽ, x̃) = g ◦ (v, x) ∈ S that
respects the group’s structure, as follows:26– e ◦ (v, x) = (v, x) for the identity element e of G
– (gg 0 ) ◦ (v, x) = g ◦ (g 0 ◦ (v, x)) for any g, g 0 ∈ G.
Now consider the set E of all contractual equilibria (C, L). Each contractual equilibrium is associated with an initial state (v, x).
To define a left action group on E, we associate for each g ∈ G and (C, L) ∈ E, a new contractual
equilibrium Φg (C, L) that respects the structure of the group, as follows:
Definition 2 The mapping from Φ : G × E → E defined by (g, (C, L)) 7→ Φg (C, L) is a transformation group if it satisfies the following axioms:
Axiom 1. For any (C, L) ∈ E, Φe (C, L) = (C, L).
Axiom 2. For all g, g 0 ∈ G and (C, L) ∈ E, Φg0 (Φg (C, L)) = Φg0 g (C, L).
Axiom 3. For any g ∈ G and (C, L) ∈ E starting from state (v, x), the contractual equilibrium Φg (C, L) starts in state (ṽ, x̃) = g ◦ (v, x).
Intuitively, a transformation group describes analogies that a principal can use to compare equilibria
across states. The definition applies equally well to equilibria to discrete-time and continuous-time
settings.
Axioms 1 and 2 are the defining properties of a left group action: Axiom 1 says that if the state isunchanged (i.e., the group identity e is applied), then the transformation is the identity mapping
over E, and Axiom 2 is an associativity axiom that will be used shortly, applied to group elementsthat are inverse of each other. Axiom 3 requires that the transformed contractual equilibria be
consistent with the effect that the group has on the state: in the translation example above, forinstance, it means that if g transforms the state v into ṽ when g operates on the state space S,
then g should also transform any contractual equilibrium that gives expected utility v to the agent
into one that gives him expected utility ṽ.
26

These two conditions define the left action. The group G must satisfy the standard definition of a group, which

include the existence of an identity element and an inverse for each element of the group.

17

4.3

State Consistency for Truthful Contracts

We begin the analysis by studying state consistency for truthful equilibria, defined as follows.
Definition 3 A contractual equilibrium (C, L) is truthful if L ≡ 0. A contract C is truthful if(C, L ≡ 0) is a contractual equilibrium.
Truthful contracts are a good starting point for the analysis of renegotiation because the principal
unambiguously knows the current cash-flow level and the agent’s continuation utility, and can use
this information to renegotiate the contract. Section 8 shows that the focus on truthful contracts
is without loss of generality as long as the strategy of the agent satisfies one of the following
regularity properties: it is either locally Lipschitz with respect to the true cash-flow process (aproperty satisfied, e.g., by truthtelling), or it responds with an arbitrarily small lag to news arrival.
We define a concept of state consistency, according to which challengers are obtained throughthe transformation group Φ. Given any contractual equilibrium (C, L), let Π(C, L) denote theprincipal’s expected payoff. Although we focus for now on truthful equilibria, we emphasize thatthe concepts in this section apply to all contractual equilibria.
Definition 4 A contractual equilibrium (C, L) starting from state (v, x) is state consistent withrespect to Φ if, after any history leading to some state (ṽ, x̃) = g◦(v, x) and continuation contractual
equilibrium (C̃, L̃), we haveΠ(C̃, L̃) ≥ Π (Φg (C, L)) ,and, reciprocally,Π(C, L) ≥ Π Φg−1 (C̃, L̃) .
State consistency entails two requirements: first, each continuation of (C, L) must sustain thecomparison with the transformation of (C, L) to the state corresponding to this continuation.27
Second, the initial contractual equilibrium must sustain the comparison with each continuation
equilibrium transformed back to the initial state, i.e., corresponding to the group element g −1 ,since g −1 ◦ (g ◦ (v, x)) = e ◦ (v, x) = (v, x).
A first observation is that a state-consistent contractual equilibrium is internally consistent as longas Φ satisfies the following monotonicity condition.
Definition 5 The transformation group Φ is monotone if for any two contractual equilibria (C, L)
and (C 0 , L0 ) starting from some common state (v, x) and giving payoffs Π(C, L) ≤ (<)Π(C 0 , L0 ) to27

This definition does not distinguish between on-path and off-path continuations. This distinction is irrelevant forthe setting of Section 3 since the agent’s lies are absolutely continuous and the report process is always on path.

18

the principal, and any g ∈ G, the principal’s payoffs in the transformed contractual equilibria(C̃, L̃) = Φg (C, L) and (C̃ 0 , L̃0 ) = Φg (C 0 , L0 ) satisfy Π(C̃, L̃) ≤ (<)Π(C̃ 0 , L̃0 ).
Monotonicity means that the ranking of the principal’s payoffs is preserved under transformations
to a different state.
Proposition 5 Suppose that Φ is monotone and that the contractual equilibrium (C, L) startingfrom state (v, x) is state consistent with respect to Φ. Then, after any finite history leading tostate (ṽ, x̃) = g ◦ (v, x), the continuation payoff for the principal is equal to her initial payoff in thecontractual equilibrium Φg (C, L).
This proposition, proved in the Appendix, implies the following result.
Corollary 1 If Φ is monotone, any state-consistent equilibrium is internally consistent.
The proof is immediate: if (C, L) is state consistent, Proposition 5 implies that any two histories
leading to the same state (ṽ, x̃) = g ◦ (v, x) give the same continuation payoff π̃ to the principal
and, hence that the players’ continuation payoffs (ṽ, π̃) of both parties are not Pareto ranked.
Strength, Stability, and Uniqueness of the State-Consistency Concept: In principle, there
may be several transformations groups to consider. The group G operating on the state space istransitive (in the group theoretic sense) if for any states (v, x) and (v 0 , x0 ) there exists some g ∈ Gsuch that (v 0 , x0 ) = g ◦ (v, x). This property is satisfied by the group constructed in our main
application (Section 5). If Φ is defined with respect to a transitive group G, it means that we cancompare contractual equilibria across any two states.
The theory of state consistency does not rely on transitivity: it is well defined for any group G.
This observation is useful if, for instance, in settings for which it is hard to establish comparisonsacross all pairs. The next proposition establishes two results pertaining to non-transitive groups.
Given a group G operating on S and a transformation group Φ defined with respect to G, we canconsider, for any subgroup G0 of G the transformation group Φ0 that is the restriction of Φ withrespect to G0 . A particular subgroup is the trivial group that contains only the identity element
e. With respect to the trivial group, Φ reduces to the identity transformation over contractual
equilibria. We have the following result.
Proposition 6 (i) Suppose that Φ is a transformation group with respect to G and that G0 is a
subgroup of G. A contractual equilibrium is state consistent with respect to Φ only if it is state
consistent with respect to Φ0 . (ii) If G0 is the trivial group, a contractual equilibrium is state
consistent with respect to Φ0 if and only if it is internally consistent.
19Part (i), whose proof is straightforward, establishes a nestedness condition for concepts of stateconsistency. Regarding Part (ii), we note that the identity transformation group is clearly monotonic, so internal consistency follows from Corollary 1. The reverse direction follows immediatelyfrom the definition of state consistency.
From Proposition 6, the stronger concepts of state consistency are obtained for transitive groups.
Fixing a group G, there may a priori exists multiple transformation groups with respect to G,
which could potentially lead to different solution concepts for state consistency. However, the next
result shows that, for a contractual equilibrium to sustain comparisons with respect to several
monotone transformation groups, these groups must, taken individually, yield the same concept ofstate consistency. Consider two monotone transformation groups Φ and Φ̃. Given a contractual
equilibrium (C, L) starting from (v, x), suppose that there is a state (v 0 , x0 ) = g ◦ (v, x) for which
the groups yield different payoffs: Π(Φg (C, L)) > Π(Φ̃g (C, L)). Applying the transformations again
from (v 0 , x0 ) to (v, x) = g −1 (v 0 , x0 ) and using monotonicity, we getΠ(Φ̃g−1 (Φg (C, L))) > Π(C, L).
Therefore, (C, L) is dominated (from the principal’s perspective) by a simple composition of elements in the two groups. In such a case, we will say that (C, L) is unstable with respect to (Φ, Φ̃),otherwise, (C, L) is called stable with respect to (Φ, Φ̃). Intuitively, instability implies that the principal’s ability to consider transformations that yield different payoffs after a given history prevents
the existence of a state-consistent contractual equilibrium.28
Proposition 7 (Concept Equivalence for Stable Contractual Equilibria) Let Φ andΦ̃ be two monotone transformation groups with respect G and (C, L) be a contract equilibrium
starting from state (v, x) that is stable with respect to (Φ, Φ̃). Then, (C, L) is state consistent withrespect to Φ if and only if it is state consistent with respect to Φ̃. Moreover, the principal’s payoffs
under the Φ and Φ̃ transformations of (C, L) to any other state (v 0 , x0 ) are identical.

5

Transformation Group: Explicit Construction

This section constructs a transitive transformation group Φ explicitly for the setting of Section 3.
28

For the setting of Section 3, it is possible to build somewhat “pathological” transformation groups different fromthose constructed in Section 5, for which state consistent contracts (according to the original transformation groupof that section) is not stable. However, unlike the original transformation group of Section 5, these transformationgroups have two potential issues: (i) they are not in the spirit of internal consistency, because they transform a given
contract into contracts that are not easily obtained from the original contract, and (ii) there need exist any truthfulcontract that is state consistent with respect to these other transformation groups.

20

5.1

Transforming Contracts Across Cash Flow Levels

Given a contactual equilibrium (C, L) starting from state (v, x), and any x̂ 6= x, we construct a
new contractual equilibrium that starts from (v, x̂). Since C is adapted to the filtration FY of theagent’s report process Y , it can be expressed as Ct = C(Ys : s ≤ t) for some functional C (fornotational simplicity, C is defined on paths of Y of varying time lengths, which implicitly allows C
to depend t).
Now suppose that the initial cash flow level is x̂ and that the agent uses strategy L. This strategy
generates a report process Ŷ , which is given by equation (7) and initial condition Ŷ0 = x̂ and thetrue cash flow process X̂ satisfies (6) subject to initial condition X̂0 = x̂. In this situation, theprincipal can choose to give the agent the consumption process Ĉt = C(Ỹs : s ≤ t), where Ỹ isconstructed from Ŷ as follows: Ỹ0 = x anddỸt = dŶt − (ξ − λŶt )dt + (ξ − λỸt )dt.

(10)

Intuitively, the process Ỹ is the report process that the agent would have produced if the initial cash
flow had been x instead of x̂ and the agent had followed the same strategy L that generates Ŷ whenstarting from x̂. We call Ỹ the virtual report process. Notice that the principal can always computethe virtual report process Ỹ from observing the report process Ŷ , thanks to equation (10).
Proposing contract Ĉ, which is based on Ỹ , allows the principal to make the consumption process
independent of the initial cash flow level x̂. By insulating the agent’s compensation process fromthe initial cash flow, the principal separates the agent’s lying incentives from the initial cash flow.
This separation works when the report process Y has a linear dynamic, as shown the next result.
Proposition 8 Suppose that the contract C defined by Ct = C(Ys : s ≤ t) together with strategy L
forms a contractual equilibrium starting from (v, x). Then, the contract Ĉ defined by Ĉt = C(Ỹs :s ≤ t) together with strategy L forms a contractual equilibrium starting from (v, x̂).
Proposition 8 allows us define the first part of the transformation Φ: we set for any x̂ ∈ R
Φ(1,x̂−x) (C, L) = (Ĉ, L).
The reason for the “1” subscript has to do with how the group G operating on the state space will
be defined and will become clear in the next subsection.
We emphasize that this transformation applies for general utility functions of the agent, as was in
the case in Proposition 2. Nothing in the proof of Proposition 8 requires that the agent have an
exponential utility function. The key to providing Proposition 8 is that the cash flow has a linear
dynamic equation (specifically, the drift of the cash flow is affine in its current value).
21

5.2

Transforming Contracts Across Promised-Utility Levels

From now on, we assume that the agent’s utility function is given by u(c) = − exp(−θc) for somerisk aversion parameter θ > 0. In particular, the agent’s flow utility u(Ct − Gt ) is always negative
and, hence, so are his promised and continuation utilities at all times.
Consider a contractual equilibrium (C, L) starting from state (v0 , x), with v0 < 0, and consider any
alternative utility level v1 = βv0 for some β ∈ (0, ∞). We define a new contract Ĉ as follows:Ĉt = Ct −log(β)
θfor all t. We have the following result:
Proposition 9 (C, L) is a contractual equilibrium starting from (v0 , x) if and only if (Ĉ, L) is acontractual equilibrium starting from (v1 , x).
Proof. Let V (C, L) denote the agent’s expected utility when he follows strategy L given contract
C. Since L is optimal for the agent, we have
V (C, L) = v0 ≥ v(C, L0 )for all L0 . For any L̂0 , let L0 = L̂0 . Then, it is straightforward to check from (8) that
V (C, L̂0 ) = βV (C, L0 ) ≤ βV (C, L) = βv0 = v1
and the inequality is tight if L̂0 ≡ L, which shows that L is optimal given Ĉ and provides utility
βv0 to the agent .

For any contractual equilibrium (C, L) and β > 0, we setΦ(β,0) (C, L) = (Ĉ, L).

5.3

Action Group

Since the agent’s flow utility is negative, the relevant state space for the agent is S = (−∞, 0) × R:
it consists of all possible pairs (v, x) of expected utility and cash flows.
Let G = (0, +∞) × R denote the algebraic group defined by the binary operation (β, δ)(β 0 , δ 0 ) =(ββ 0 , δ + δ 0 ) for all (β, δ), (β 0 , δ 0 ) ∈ G. It is straightforward to check that G is an Abelian groupwith identity element (1, 0).
Moreover, the operation (β, δ) ◦ (v, x) = (βv, x + δ) defines a left action of G on the state space S:
the associativity and identity axioms are straightforward to check.
22

Finally, the group G defines a left action on the set of contractual equilibria, as follows:
1. Sections 5.1 and 5.2 define Φ(β,0) (C, L) and Φ(1,δ) (C, L) for all β > 0, δ ∈ R and contractual
equilibrium (C, L).
2. For any (β, δ) ∈ G, we define Φ(β,δ) (C, L) by
Φ(β,δ) (C, L) = Φ(β,0) (Φ(1,δ) (C, L))
3. It is straightforward to check that this definition is consistent with 1. and that Φ satisfies
Axioms 1,2,3 of Definition 2. Hence, Φ is a transformation group.29
4. Finally, it is also straightforward to check that Φ is monotone: if Π(C, L) ≥ Π(C 0 , L0 ) for any
two contractual equilibria starting from the same state, then the constructions of Sections 5.1
and 5.2 preserve this payoff relation.
These observations are summarized by the following proposition.
Proposition 10 The mapping Φ : G × E → E is a monotone transformation group.

6

Characterization of Truthful Renegotiation-Proof Contracts

This section derives a functional form for the payoff of the principal in any truthful state-consistentcontract. It then introduces a mild strengthening of state consistency, in which the set of challengers is convexified. Renegotiation-proof contracts are those that sustain the comparison withthis convexified set of challengers. The convexification, together with the strict concavity of theagent’s utility function, implies that the contractual variables themselves, and not just the continuation payoff of the principal, are Markovian, and leads to a complete characterization of truthful,renegotiation-proof contracts. The class of truthful, renegotiation-proof contracts is parametrized
by a single real number, which may be interpreted as the sensitivity of the agent’s continuation
utility to his reports.
Truthful Contracts and Promised Utility:
Given a truthful contract C, the filtrations FX and FY are identical and the agent’s continuation
utility at time t isZ ∞Wt = E−r(τ −t)e(u(Cτ )) dτ

FtY


.

t
29

To prove associativity, one must show that consumption translation commutes with the virtual cash flow con-struction of Section 5.1, which is immediate from the construction.

23

The quantity Wt is the agent’s promised utility at time t : it is the agent’s expected discounted
utility if he truthfully reports his cash flows. Since the promised utility process W = {Wt }t≥0 is
adapted to the filtration FY , the Martingale Representation Theorem implies that W satisfies the
dynamic equation30
dWt = (rWt − u(Ct ))dt + σSt dB̃t(11)where the process St is FY -adapted and B̃t is an FY -adapted Brownian motion under the probability
measure in which the agent reports truthfully, i.e.,dB̃t =

dYt − (ξ − λYt )dt.
σIf the agent does not report his cash flow truthfully, B̃ is no longer a martingale: plugging Equation (7) into the previous equation yields
dB̃t =λ(Yt − Xt )dt + Lt dt + σdBt
.
σ

(12)

Intuitively, the cash flow shocks reported by the agent’s are biased upward if (i) the agent overreports his cash flow (Lt > 0) or (ii) the actual cash flow is lower than the reported cash flow
(Xt < Yt ). Effect (i) is straightforward. Effect (ii) comes from mean-reversion: if Xt < Yt , thenXt has a higher mean than what the principal expects based on Yt . Even if the agent does notproduce any additional lie (Lt = 0), reported increments are positively biased.
It follows from (12) that when the agent lies, his promised utility evolves as
dWt = (rWt − u(Ct ))dt + St (Lt dt + λGt dt + σdBt ),(13)

where {Bt }t≥0 is the standard Brownian motion. The coefficient St is the sensitivity of the agent’s
promised utility to the agent’s report increment at time t. It will be treated as a choice variable ofthe principal in the recursive formulation of the problem.
Agent’s Incentives: Persistence of the agent’s private information implies that past lies can havea long term impact on incentives: if the agent has lied even for a short period before time t, he has
affected the report history Y t = {Ys }s≤t and therefore also affected his future consumption flow C
and his future incentives to report the truth. Intuitively, if the agent underreports his cash flow
increment dXt at time t, he affects his utility through two channels. First, this reduces his promised
utility, which depends on the report dYt by a sensitivity factor St , and thus reduces the consumption
stream that the principal aims to give the agent. Second, to deliver a given level of promised utility,
the principal must provide higher transfers to the agent if the agent generates lower cash flows. The
first channel incentivizes the agent to report higher cash flows and the second channel incentivizes
30

See, e.g., Karatzas and Shreve (1991).

24

him to report lower cash flows. For the contract to be truthful, these two incentives must balanceeach other, at least on the equilibrium path. The agent’s reporting incentives are the subject ofSection 9.

6.1

Structure of the Principal’s Payoff

Consider a truthful, state-consistent contract C for the transformation group Φ constructed in
Section 5. Since the contract is truthful, there is no difference on the equilibrium path between Xt
and Yt , or between Vt and Wt .
We will use (w, y) instead of (v, x) to denote the state, to make these equalities clear. For any state(w, y) ∈ (−∞, 0) × R, let Π(w, y) denote the principal’s expected payoff for the transformationof (C, 0) that corresponds to state (w, y). From Proposition 5, Π(w, y) is also the principal’s
continuation payoff after any history leading to state (w, y). We now derive the functional form of
Π(w, y).
First, consider the dependence on the cash flow level y. As noted in Section 5.1, the only differencein continuation payoffs for the principal between histories starting from state (w, y) and (w, y 0 )R∞concerns the expected discounted transfer from the agent Υ(y) = E[ 0 e−rt Yt dt|w, y], because theprocesses C have the same distribution independently of y.
To compute Υ(y), note that since the contract is truthful, we have Yt ≡ Xt . Moreover, the cash
flow process Xt has an explicit formula, which is computed in the Appendix (Equation (50)), theexpected discounted transfer from the agent satisfies

Z t

Z ∞
−rt
−λt
λ(s−t)
Υ(y) =
ee y+E
e
ξds dt.
00After simplification, this yields
Υ(y) =

ξy+.
r + λ r(r + λ)

(14)

y
.
r+λ

(15)

This yields the following result:Π(w, y) = Π(w, 0) +Next, consider two different promised utility levels w0 and w1 = βw0 for the agent, where β > 0.
The translation performed in Section 5.2 implies thatΠ(βw0 , y) =log(β)+ Π(w0 , y).
θr

The previous analysis yields the following result.
25

Proposition 11 For any truthful, state consistent contract C, the principal’s payoff satisfies thefollowing relation:
Π(w0 , y 0 ) =(y 0 − y) log(w0 /w)++ Π(w, y).
r+λrθ

In particular, the principal’s continuation payoff in state (w, y) satisfies
Π(w, y) =ylog(−w)++ Π(−1, 0).
r+λ
θr

(16)

Proposition 11 has an immediate but important corollary:
Corollary 2 If C is truthful and state consistent, the principal’s continuation payoff takes theform Πt = Π(Wt , Yt ) where Π(·, ·) is twice continuously differentiable.

6.2

Convexification and Renegotiation-Proof Contracts

Proposition 11 established that the continuation payoff of the principal in any truthful, stateconsistent contract is Markov in the state variables (Wt , Yt ). In principle, the contract itself neednot be Markov: there could be multiple continuation contracts starting from a given state (w, y)that give the principal the same payoff, and all of these continuations could in principle be used
after various histories.
In order to guarantee the contract is Markov, we impose a mild strengthening of state consistency,
in which the set of challengers to a given continuation is the convex hull of the relevant set ofcontract transformations.
When the agent’s utility is concave and a contractual equilibrium has multiple continuations giving
the agent the same expected utility, it is natural for the principal to consider convexifications ofthese continuations, as they may be cheaper.
Specifically, consider two truthful contracts C 1 , C 2 starting from the same cash flow x and giving
the same utility v to the agent. For any λ ∈ [0, 1], consider the contract C λ defined as follows: forany t and report history {Ys : s ≤ t},u(Ctλ ) = λu(Ct1 ) + (1 − λ)u(Ct2 ).

(17)

The consumption level Ctλ is uniquely defined if u is strictly increasing and continuous, which is thecase for exponential utility functions. In words, for any t and report history {Ys }s≤t , Ctλ provides
a flow utility to the agent that is a fixed convex combination of the flow utilities that he gets underCt1 and Ct2 . If C λ is truthful, this implies that it provides expected utility v to the agent because(i) C 1 and C 2 both give expected utility v to the agent, and (ii) all three contracts generate the26same distribution for the report process Y . The next result shows that C λ is indeed truthful andformalizes this result.
Proposition 12 Suppose that u(c) = − exp(−θc) for some parameter θ > 0. For any λ ∈ [0, 1]
and initial cash flow x, the contract C λ is truthful and gives the agent expected utility v.
Given a truthful contract C and state (v, x), let K(C) denote the set of all continuation contractsof C, and letnK(v, x) = Φg (C̃) : (g, C̃) ∈ G × K(C)os.t. g ◦ (vC̃ , xC̃ ) = (v, x)

where (vC̃ , xC̃ ) denotes the starting state of contract C̃. In words, K(v, x) consists of all contracts
starting from state (v, x) that are transformations of continuation contracts of C.
Finally, letConv(K(v, x)) = {C λ : λ ∈ [0, 1], C 1 , C 2 ∈ K(v, x)}
denote the set of convex combinations of contracts in K(v, x).
The set Conv(K(v, x)) defines the class of all challengers that the principal can consider, after any
history leading up to state (v, x), to replace the current continuation contract.
Definition 6 A truthful contract C is renegotiation-proof if (i) the pair (C, L ≡ 0) forms a stateconsistent contractual equilibrium and (ii) for any history leading to state (v, x), the continuation
contract C̃ satisfies
Π(C̃) ≥ Π(C 0 )(18)

for all C 0 ∈ Conv(K(v, x)).
This definition captures the following intuition: if the principal can come up with some challengers ofa contract, possibly by comparing contract continuations across states, then she can also consider
the convex combination of these challengers. Convexification is used in Lemma 1 to pin down
contractual variables as a function of the state (Proposition 13).

6.3

Contractual Variables

We now derive the functional form of contractual variables Ct , St for all renegotiation-proof contracts.
Proposition 13 For any truthful, renegotiation-proof contract, there exist constants c1 , s̄ such that
Ct = c1 −log(−Wt )
θ
27

a.s.

(19)

St = −Wt s̄

a.s.

(20)

for all t ≥ 0.
The parameters c1 and s̄ are the consumption and sensitivity levels provided at time 0 by thetransformation of the contract that starts with promised utility −1 (any cash flow level).
From Proposition 13, the parameters c1 and s̄ fully characterize contract C. We will see in Section 9that a contract to be truthful, c1 and s̄ must satisfy an additional equation, which will further reduce
the set of renegotiation-proof contracts to a single parameter family.
Proof. From (11), the statement of Proposition 13 is equivalent to W being a geometric Brownian
motion or, equivalently, to the Itô process Z given by Zt = log(−Wt ) having constant drift andvolatility.
To prove that the drift and volatility of Z are constant, we will use an intermediate result. Let (w, y)denote the initial state of the truthful renegotiation-proof contract C. Fix a time T and history up
to time T and let C̃ and (w̃, ỹ) denote the continuation contract and state following this history.
Let Ĉ denote the transformation of C̃ that starts in state (w, y): letting g denote the element of G
such that (w̃, ỹ) = g ◦ (w, y) (which is always possible since G is transitive) we let Ĉ = Φg−1 (C̃).
The contract C̃ is truthful as a continuation of C, which is truthful. Therefore, Ĉ is also truthful
because Φ preserves truthfulness (Propositions 8 and 9). The contract Ĉ may be proposed by theprincipal at time 0, instead of contract C, as it starts from the same state (w, y). We now show thatthe stochastic processes C = (Ct )t≥0 and Ĉ = (Ĉt )t≥0 must be identical functions of the realized
cash-flow process Y = (Ys )s≥0 . This result, which is proved in Appendix B, is formalized as follows.
R∞
Let L denote the vector space of processes Z such that E[ 0 e−rt |Zt |dt] < ∞ endowed with theR∞norm kZkL = E[ 0 e−rt |Zt |dt]. By assumption, any contract C belongs to L under truth-telling.
Lemma 1 The processes C and Ĉ are identical in L: kC − ĈkL = 0.
Let {Ŵt }t≥0 denote the agent’s promised utility under contract Ĉ. Ŵ is an FY -adapted Itô process
given byZ ∞Ŵt = E−r(s−t)eu(Ĉs )dsYz : z ≤ t .

(21)

t

Lemma 1 implies that Ŵ is identical to W and that the process Ẑ : {Ẑt = log(−Ŵt )}t≥0 is identicalto the process Z : {Zt = log(−Wt )}t≥0 .
Let µt and σt denote the drift and volatility of Z at time t. Our objective is to show that µt andσt are constant. Since Ẑ and Z are identical, we have:dẐt = dZt = µt dt + σt dBt .
28

By definition of Φ, the contract Ĉ = Φg−1 (C̃) is obtained by translating the process C̃ = {CT +t }t≥0
by the constant 1θ (log(−WT ) − log(−w)) (setting β = log(−Wt / − w) in Section 5.2) and making
it independent of the initial condition YT , as explained in Section 5.1. The latter condition isequivalent to the requirement that Ĉ depends on the Brownian path {Bt }t≥0 in the same way thatC̃ depends on the Brownian path {BT +t − BT }t≥0 .31
From (21) and the fact that u(c) = − exp(−θc), this implies that the process {Ẑt }t≥0 is identicalin law to the process Z̄ = {ZT +t − ZT + Z0 }t≥0 conditioned on FTY . The latter process has drift{µT +t }t≥0 and volatility {σT +t }t≥0 . Equality in law of Ẑ and Z̄ implies that the drift and volatilityof Ẑ at time zero is equal to the drift and volatility of Z̄ at time t = 0, i.e.,µ0 = µT

σ0 = σT .

Since the time T and history up to time T were arbitrary, this shows that µT and σT are constantand proves the proposition.
Proposition 13 immediately implies the following corollary:
Corollary 3 Any truthful, renegotiation-proof contract is Markovian: there exist functions c(·)
and s(·) such that Ct = c(Wt , Yt ) and St = s(Wt , Yt ) for all t.
In fact, Proposition 13 shows a stronger result: the functions c(·, ·) and s(·, ·) are independent of
Yt : the agent’s consumption at any time depends only on his current promised utility, and so does
the sensitivity St .32
Proposition 13 provides a necessary condition for the form of truthful renegotiation-proof contracts.
This necessary condition is strong, as it reduces the family of such contracts to a two-parameterfamily with parameters c1 and s̄. This reduction to two parameters is essentially due to state
consistency and convexification.33
31

With a truthful contract starting from cash flow y, the reported cash flow process Y has an explicit formula,
Rt
Rt
given by Yt = e−λt y + 0 eλ(s−t) ξds + 0 eλ(s−t) σdBs (see Equation (50)). Therefore, the path of Y is entirely pinned
down by the initial condition y and the path of B. Independence of the contract with respect to the initial conditiony then shows the claim. Remark: although easier to understand with the closed form of Y , the claim holds as longas the cash flow process Y solves an SDE with noise B that has a unique solution (see, e.g., Cherny (2002, Theorem3.2)).
32
For readers familiar with the concept of symmetry in physics and Noether’s theorem, this property may be viewed
as a conservation law of the contract that stems from state consistency: state consistency with respect to Yt may beviewed as a symmetry of the contract expressed in terms of the left group action C 7→ Φ(1,δy) (C) and invariance ofthe contract with respect to Yt is a consequence of this symmetry.
33
From an algebraic perspective, each truthful Markovian contract generates an orbit in the set T of all truthful29

In fact, the requirement that truthtelling be incentive compatible for the agent imposes a relationship between the variables c1 and s̄, as follows. Let u1 = u(c1 ). The following result is shown in
Section 9.
Theorem 1 The set of truthful renegotiation-proof contracts is characterized by Equations (11),(19), and (20), and u1 and s̄ ∈ (0, θ) satisfy the following relation:
u1 = −s̄λ.
θ − s̄

(22)

Theorem 1 shows that each truthful renegotiation-proof contract can be reduced to a single parameter, s̄. To optimize the principal’s payoff with respect to s̄, the next proposition computes thepayoff explicitly.
Proposition 14 The principal’s payoff is given by:




1
1 2 2
1
log(−u1 (s̄)) +r + u1 (s̄) − σ s̄ + D(w, y)
θr
r
2

(23)

s̄λ
where u1 (s̄) = − θ−s̄andD(w, y) = log(−w) +7θ(ry + ξ).
r+λComparative Statics

The following comparative statics obtain by maximizing (23) with respect to s̄.

7.1

Persistence

The first result is to show that the sensitivity of the contract, s̄ is increasing in the persistence of
the agent’s type.
Proposition 15 The optimal sensitivity s̄ is decreasing in λ.
Intuitively, when the cash flow is more persistent (λ is lower), the agent’s current report has morebearing on expected future cash flows. This raises the magnitude of the agency problem andrequires a higher a sensitivity to induce truthful reporting.
Markovian contracts through the left action of group Φ. Proposition 13 says that the quotient of T with respect tothese orbits (or equivalence classes) is two-dimensional.

30

The proof of Proposition 15 is straightforward. Ignoring terms and factors in the principal’s objective (23) that do not influence the choice of s̄, the optimal choice of s̄ is the solution to


1
1 2 2
log(−u1 (s̄)) +u1 (s̄) − σ s̄ ,
r
2
where
u1 (s̄) =

(24)

−s̄λ
.
θ − s̄

The function s̄ 7→ u1 (s̄) is submodular in (s̄, λ), as is easily checked. Therefore, the second termof (24) is submodular. The logarithmic term breaks up into separate functions of s̄ and λ, and istherefore modular in (s̄, λ). Submodularity of the objective implies that s̄ is decreasing in λ (seeTopkis 1978). The intuition here can be back traced to the incentive compatibility condition (32):s̄ =θ(−u1 ).
λ − u1

A lower λ (higher persistence) implies a higher sensitivity s̄.

7.2

Impact of Volatility and Discount Rate

It is easy to check from (23) that the principal’s payoff is decreasing in the volatility of the agent’s
cash flow. The next result shows that the optimal sensitivity coefficient is also decreasing in cash
flow volatility.
Proposition 16 The optimal sensitivity s̄ is decreasing in σ.
Proof. Expression (23) is submodular in σ and s̄. The result follows from Topkis (1978).
This result is intuitive: A higher volatility exposes the agent to more risk, other things equal. Since
the agent is risk averse, it is optimal for the principal to offset this by reducing the agent’s exposure
to exogenous cash flow fluctuations, by reducing s̄. Perhaps more surprisingly, the sensitivity isdecreasing in the patience of the players, as shown in the Appendix.
Proposition 17 The optimal sensitivity s̄ is increasing in r.
A possible intuition is that a less patient agent is more concerned about getting an immediate
subsidy (or tax reduction) than about his future utility. Inducing truth-telling thus requires ahigher sensitivity of the agent’s future utility to his current report.

31

7.3

Long-Run Properties

Bloedel, Krishna, and Strulovici (2020) show that the agent’s utility always has a negative drift
under the optimal contract, but that the long-run properties of consumption and promised utility
depend on the parameters of the agent’s cash flow process. Fixing the volatility parameter, theoptimal contract induces immiserisation when persistence is low, but sends the agent to bliss when
persistence is high. Fixing the degree of persistence, the optimal contract induces immiserisation
when volatility is high, but sends the agent to bliss when volatility is low.

8

Observability Theorem and Revelation Principle

Reconstructing the Agent’s True Cash Flow: An Observability Theorem
We focused up to now on truthful equilibria. Consider instead a contractual equilibrium in which
the agent lies on path about his cash flow. We will see that, given an equilibrium strategy L thatis FX adapted, the principal can under mild regularity conditions on L reconstruct the true cash
flow Xt and the agent’s true continuation utility Vt from the agent’s report process Y . To this end,
notice that the agent’s lying process is FX -adapted and may thus be expressed as a functional
Lt = L(t, Xs : s ≤ t),(25)

which depends, at each time t on the path of X until time t. Given the functionals {L(t, ·)}t≥0 —which the principal knows in equilibrium—and the report process Y , we can construct a processX̃t defined by the equationdX̃t = dYt − L(t, X̃s : s ≤ t)dt

(26)

and the initial condition X̃0 = Y0 = X0 . Equation (26) is a functional stochastic differential
equation in which the unknown is the process X̃ and the primitive is the process Y . From (7),the true cash flow process X is a weak solution of (26). Moreover, any strong solution of (26) is,by definition, adapted to Y . If (26) has a unique solution, the principal can back out the process
X by observing Y and using her knowledge of the agent’s equilibrium strategy. This intuition isformalized by Theorem 2, which is proved in Appendix C.34
Theorem 2 (Observability Theorem) Consider a contractual equilibrium (C, L) such that L
is locally bounded: for each T > 0, there exists M (T ) > 0 such that supt≤T |Lt | ≤ M (T ) a.s. Then,the following holds:
34

While the model’s horizon is infinite, Theorem 2 provides conditions under which, for each T > 0, the agent’s

reports up to time T reveal the agent’s continuation value and the current-cash flow at time T . Thus, to prove
Theorem 2, it suffices to show that it holds for any finite horizon [0, T ], where T > 0 is arbitrary.

32

1. If Equation (26) has a strong solution, then this solution is unique, and X is FY -adapted.
2. If L is locally Lipschitz continuous,35 then Equation (26) has a unique strong solution and X
is FY -adapted.
3. If there exists some ε > 0 such that for each t > 0, the functional L(t, ·) is independent of{Xs }s∈[t−ε,t] , then Equation (26) has a unique strong solution and X is FY -adapted.
Conditions given in 2. and 3. both include truthtelling as a feasible strategy. In particular, thetruthful strategy L ≡ 0 is constant and thus clearly Lipschitz. The lag ε appearing in 3. may beviewed as small behavioral constraint or frictions which requires that agent’s response time to new
information is no exactly zero, but bounded below by some arbitrarily small constant.36
To interpret Theorem 2, notice that the principal’s belief about the agent’s cash flow may be quite
different from the report made by the agent. Given some contract C, the agent can affect histransfers by overreporting or underreporting his cash flows: these reports mechanically feed into
the transfer specified by C, regardless of their effect on the principal’s belief. Theorem 2 says thatthe principal’s belief process matches the true cash flow as long as the principal’s belief equation (26)
has a strong solution (Part 1), which is guaranteed if the agent’s strategy exhibits some arbitrarily
small lag (Part 3) or if it is Lipschitz for some arbitrary Lipschitz constant (Part 2).
This distinction is conceptually important in the context of renegotiation: it means that in equilibrium the principal knows the agent’s continuation utility under the current contract when renegotiating with the agent, even if the agent has fed inaccurate reports into the contract C. Thus,
it is legitimate to treat the actual cash flow, Xt , and agent continuation value Vt as state variables
for the purpose of analyzing renegotiation, as long as the agent’s strategy satisfies the assumptionsof Theorem 2, which we impose as part of the admissibility requirement for L.
Focus on Adapted Strategies
The assumption that the agent’s lying process is adapted to FX plays a key role in Theorem 2.
While the focus on adapted strategies is quasi-universal in economic models with diffusion processes, this focus is generally not innocuous. In particular, it rules out mixed strategies and it is apriori with loss of generality. In diffusion models, there have been very few attempts to model mixed
35

This means that for each T > 0, there exists L̄(T ) > 0 such that all functionals {L(t, ·)}t≤T are L̄(T )-Lipschitz

continuous over their respective domain.
36
The truthful renegotiation-proof contracts characterized by Theorem 1 are truthful even when the agent’s strategy
is allowed to violate both conditions 2. and 3. of Theorem 2, since none of these conditions is imposed in theverification that truthful behavior is optimal in Section 9.2. Imposing condition 2. or 3. is useful only to guaranteethat the focus on truthful contracts is without loss of generality, by Theorem 3.

33strategies in general as this raises difficult technical problems.37 Focusing on adapted strategies isa priori restrictive, but for diffusion models there are compelling reasons for this focus, that are
independent of any concern for renegotiation or private information. For example, the Martingale
Representation Theorem, which is used by DeMarzo and Sannikov (2006), Sannikov (2008) andnumerous subsequent papers, relies on this assumption. Williams (2011), on which the main application of the present paper is based, assumes that agent’s reporting strategy is adapted to theobserved process (cf. last paragraph of p. 1239 in that paper).
There are separate reasons to think that mixing would be of little help in the present setting. In
continuous-time models, the principal has infinitely many opportunities to extract information fromthe agent, and one may expect any uncertainty caused by mixing would be quickly resolved as theprincipal would renegotiate contracts with arbitrarily high frequency. For example, Maestri (2017)
and Strulovici (2017) study related protocols in discrete time and found that as the frequency of
renegotiation becomes arbitrarily large, the private information of the agent is revealed arbitrarily
quickly. This justification lies outside the scope of the present paper as there is no explicit protocolof negotiation in the present paper.

8.1

Revelation Principle

From Theorem 2, the principal knows the agent’s true output and continuation value even incontractual equilibria that are not truthful. Therefore, it is legitimate for the principal to consider
challengers based on the true continuation state (Vt , Xt ), even if the contractual equilibrium is nottruthful.
As noted in Section 4, the transformation group and the concept of state consistency defined in that
section applies equally well to contractual equilibria as it does to truthful contractual equilibria,
and the comparisons are legitimate due to Theorem 2.
Since the concept of a state-consistent contractual equilibrium is well defined, we can ask whether
such equilibrium is outcome equivalent to a truthful contractual equilibrium, i.e., whether a versionof the Revelation Principle holds in this setting.
In the absence of commitment, it is well known that the Revelation Principle need not hold.38
However, a version of this principle holds in our setting, as described next.
Say that two contractual equilibria starting from the same state are outcome equivalent if they
induce the same net transfer after any history {Xs }s≤t of the real output.
37
38

One such attempt is due to Bernard (2016)’s PhD thesis, which focuses on this question.
See, e.g., Bester and Strausz (2001).

34

Theorem 3 Suppose that Φ is monotone. Then, any state-consistent contractual equilibrium isoutcome-equivalent to a truthful state-consistent contractual equilibrium.
Proof.

Consider any contractual equilibrium (C, L). As with the standard Revelation Principle,

we can replace the contract C by a contract Ĉ in which the agent truthfully reports his cash flow,
and the principal simulates the lies that the agent would have made in the initial equilibrium andmakes the same net transfer to the agent that she would have made in the initial equilibrium. By
construction, the truthful contractual equilibrium (Ĉ, L̂ ≡ 0) is outcome-equivalent to the initial
contractual equilibrium.
We must show that if (C, L) is state consistent with respect to Φ, then so is (Ĉ, 0). Let (v, x)
denote the initial state (which is the same for both contractual equilibria) and consider any real
output history {Xs }s≤t leading to some state (v 0 , x0 ) = g ◦(v, x). Let (Ĉ 0 , 0) denote the continuation
equilibrium of (Ĉ, 0) following this history and (C 0 , L0 ) denote the continuation equilibrium of (C, L)
following this corresponding history.
By outcome equivalence, we haveΠ(Ĉ 0 , 0) = Π(C 0 , L0 )

(27)

and
Π(C, L) = Π(Ĉ, 0).
Since Φ is monotone, the previous equality implies thatΠ(Φg (C, L)) = Π(Φg (Ĉ, 0)).

(28)Comparing (27) and (28) and using the fact that (C, L) is state consistent then shows that (Ĉ 0 , 0)also satisfies the definition of state consistency (Definition 4).

9

Necessary and Sufficient Conditions for Incentive Compatibility

This section proves Theorem 1: it shows that (22) is necessary and sufficient for the contracts of
Proposition 13 to be incentive compatible (i.e., truthful). The argument used to prove sufficiency
illustrates a strategy, potentially useful in other problems, to deal with an unbounded reporting
domain by expending the strategy space of the agent.

9.1

Necessity

Consider any contract that has the form of Proposition 13, which is characterized by parametersc1 ∈ R and s̄ ≥ 0, such that Ct = c(Wt ) and St = s(Wt ) for all t, where the functions c(·) and s(·)
35

are given by c(w) = c1 − log(−w)/θ and s(w) = s̄(−w). The promised utility of the agent evolves
asdWt = (r + u1 )Wt dt + s(Wt )(dGt + λGt dt + σdBt )(29)

where Gt = Yt − Xt where u1 = u(c1 ). Moreover, the agent’s actual consumption is c(Wt ) −Gt . Therefore, the agent cares about Yt and Xt only through their difference Gt . The agent’s
optimization problem reduces toZ ∞v(w, g) = sup E
L

e

−rt


(u(c(Wt ) − Gt )) dt

0subject to dGt = Lt dt, G0 = g, and (29). The HJB equation for this problem is
0 = sup {u(c(w) − g) − rv(w, g)
l


1
2 2
+vw (rw − u(c(w)) + s(w)(l + λg)) + vg l + (s(w)) σ vww . (30)2
It is easy to show by applying the same controls starting from different values of w that the valuefunction has the form v(w, g) = wf (g) for some function f to determine. Incentive compatibility
obtains if one finds a solution f such that f (0) = 1, which means that the agent cannot do better
than get his promised utility w. The Bellman equation becomes, after simplification and dividing
throughout by (−w),

0 = sup u1 exp(θg) + f (g)(−u1 + s̄(` + λg)) − f 0 (g)` .

(31)

`

A priori, the function f defining the agent’s value function need not be everywhere differentiable,
but it is a viscosity supersolution of (31) (see, e.g., Øksendal and Sulem (2004)), which means inthis context that the supremum in (31) must be less than or equal to 0. Moreover, v(w, g) is clearlydecreasing in the gap g, which implies that f is increasing. This implies that f is left and rightdifferentiable at 0. Incentive compatibility implies, for g = 0, that it is optimal for the agent not
to increase the gap above zero, which requires thats̄ − fr (0) ≤ 0,
and it is optimal not to decrease it below zero, which requires thats̄ − fl (0) ≥ 0.
Therefore, incentive compatibility implies that fl (0) ≤ s̄ ≤ fr (0).
We now prove that these inequalities are tight. Suppose by contradiction that one of these inequalities is strict, for example, s̄ < fr (0). This implies that there exists a right neighborhood of 0 such36

that for g ∈ (0, η) it is strictly optimal for the agent to reduce the gap to zero, at an infinitely
negative rate. We approximate the agent’s optimal strategy by considering an arbitrarily negative
lying rate −K over some small time window, and taking the limit as K → +∞. Specifically, suppose that at time t the state is (w, g) for some fixed g ∈ (0, η), so that the agent expected utility attime t is v(w, g) = wf (g), and suppose that the agent lies at an arbitrarily large rate −K between
times t and tK = t + g/K, which brings the gap to 0 at time tK . The agent’s promised utility overthis time interval obeys the dynamic equation39
dWt = Wt (r + u1 + K s̄)dt + s̄(−Wt )σdBt ,which integrates to WtK = ŴtK exp(s̄K), where ŴtK is the agent’s promised utility at time tK if hedoesn’t lie during this period. The agent’s expected utility at time tK is therefore v(WtK , GtK ) =ŴtK exp(s̄g)f (0) since we chose tK so that GtK = 0. As K → +∞, we have tK → t, ŴtK → w,
and v(WtK , GtK ) ≤ v(w, g) since lying at the arbitrarily large rate −K is only one of many possible
strategies for the agent. Moreover, incentive compatibility requires that f (0) = 1. Taken together,
these observations yield wf (g) ≥ w exp(s̄g). Dividing by w < 0 and noting that the argument
works for all g ∈ (0, η), we get
f (g) ≤ exp(s̄g)
for all g ∈ (0, η). Since the functions on both sides of the equation are equal to 1 for g = 0, we get
fr0 (0) ≤d[exp(s̄g)]
= s̄,
dg
g=0

which yields the desired contradiction. By a similar argument, fl (0) = s̄.
In conclusion, f is differentiable at 0, with derivative s̄. This implies that the value functionv̄(w, x, y) = v(w, y − x) of the agent is differentiable with respect to x and to y whenever x = y.
In Appendix D (Lemma 2), this differentiability is used in an envelope argument to show that,necessarily,s̄(−w) = f 0 (0)(−w) = vx (w, x, x) = −vy (w, x, x) = EZ ∞e−(r+λ)t u0 (Ct + Xt − Yt )dt .

0

The right-hand side is computed explicitly in Appendix D (Lemma 3), yieldings̄ =θ(−u1 )∈ (0, θ).
λ − u1(32)

Notice that s̄ < θ for all u1 < 0 and λ > 0. Intuitively, if s̄ were higher than θ, the agent would
want to exaggerate the cash flow in order to artificially increase his promised utility. The effectof earning less than the reported cash flow would reduce the agent’s flow utility at rate θ, which
would be dominated by the increase in promised utility governed by the sensitivity parameter s̄.
Incentive compatibility rules this case out.
39

The equation is exact up to a second-order term, which becomes negligible as K → +∞.

37

9.2

Sufficiency

We have established that (22) is necessary for incentive compatibility. We now show that it issufficient. The objective (31) is linear in `, which has unbounded domain. If the contract is nottruthful, the agent therefore wants to lie at an infinite rate. To account for this, we now allow theagent to report jumps in his cash flows, which expands the agent’s strategy space.40
Let Wt+ (∆L) denote the agent’s promised utility if he reports a jump ∆L in his cash flow attime t. For contracts with a fixed sensitivity parameter, as considered here, a natural closure of
the contract to report jumps is to stipulate that
Wt+ (∆L) = exp(−s̄∆L)Wt .

(33)

To see this, notice that if the agent lies at an arbitrarily large rate K between times t and t + ε, his
promised utility satisfies, ignoring second-order effects (see the analysis of the necessary conditionin Section 9.1 for a formal argument), the dynamic equation dWt = s̄(−Wt )Kdt. This yields
Wt+ε = exp(−K s̄ε)Wt , and results in a gap change Gt+ε = Gt + Kε. Combining the last two
equations yields Wt+ε = exp(−s̄(Gt+ε − Gt ))Wt , which explains (33).
Report jumps amount to impulse controls on the part of the agent (see for example, Øksendal and
Sulem (2004)). The HJB equation (31) becomes
0 = max sup u1 exp(θg) + f (g)(−u1 + s̄(` + λg)) − f 0 (g)` ,
`∈R

)
sup

{exp(−s̄∆L)f (g + ∆L) − f (g)} . (34)

∆L∈R\{0}We now show that the function f (g) = exp(s̄g) solves this equation. With this value for f , thesecond term of the equation is always equal to zero. Therefore, it suffices to show thatsup {u1 exp(θg) + exp(s̄g) (−u1 + s̄ (` + λg)) − s̄ exp(s̄g)`} ≤ 0

(35)

`

for all g. The terms involving ` cancel out and (35) reduces to
u1 exp(θg) + exp(s̄g) (−u1 + s̄λg)
Convexity of the exponential function implies that for all g 6= 0,
exp(θg) > exp(s̄g) + exp(s̄g)(θ − s̄)g.
40

Admissibility is unchanged: it requires that the transversality condition (9) holds. This condition is used to checkin a standard verification argument that the solution of the HJB equation must be equal to the value function.

38

Since u1 < 0, this implies that (35) is satisfied if
exp(s̄g) (u1 (θ − s̄)g + s̄λg) ≤ 0.
The second factor is zero, from (22), which concludes the proof.
An optimal control associated with the Bellman equation is to set ∆L = −g if g 6= 0 and ∆L = 0
otherwise, and ` always equal to zero. It is therefore optimal for the agent to (i) always report
truthfully if he has been truthful in the past, and (ii) immediately correct any existing gap between
real and reported cash flows. It means, in particular, that if the principal did not know the initial
cash flow, the contract is still incentive compatible. This optimal control is essentially unique.
Indeed, between two impulse controls, the first term of the Bellman equation must equal zero,
which holds only if g = 0.

10Discussion

10.1

Summary: Concept of Renegotiation

This paper introduces a concept of renegotiation that strengthens internal consistency by exploiting comparisons across states. These comparisons are weaker than those imposed by strongly
renegotiation-proof equilibria (Farrell and Maskin (1989)) and could be applied to study stable“social norms” (Asheim (1991)), with the following cognitive interpretation: players may recognize relations between equilibria across different states, just as they recognize, in a repeated game,whether continuation payoffs across different histories are Pareto ranked. This requires players tothink by analogy rather than come up with radically new equilibria. From a modeling perspective, economic analysis often relies on specific payoff or dynamic structures (e.g., CARA or CRRA
preferences, linear or geometric growth). For such models, the present approach provides a natural
way to model renegotiation.
The concept is well-defined in any of the following situations: (i) there is no private information(time could be discrete or continuous), (ii) there is private information but we focus on truthfulcontracts, or (iii) there is private information, and the setting is a diffusion model and strategies
are adapted to the underlying Brownian process, as in the main application of this paper. In thislast setting, the paper provides an Observability Theorem and a Revelation Principle that address
informational asymmetries.

39

10.2

Robustness and Extensions

State Consistency in Games: The concept of state consistency developed in this paper has
focused on bilateral contractual relationships. State consistency could in principle be extended to
dynamic games with an infinite horizon. The idea is as follows. Consider a dynamic game withn agents, whose environment at time t is described by some state variable Xt ∈ X . Suppose thatgiven (i) any subgame-perfect equilibrium (SPE) E starting from state X0 = x and (ii) any otherstate x0 6= x in X , it is possible to construct another SPE E 0 = Φx,x0 (E) starting from x0 . Let
V 0 = (V10 , . . . , Vn0 ) denote the vector of payoffs of all players from equilibrium E 0 . Suppose that,
under the original SPE E, the game reaches a history at which the state is x0 , and the continuation
equilibrium is Ē 0 . Let V̄ 0 = (V̄10 , . . . , V̄n0 ) denote players’ continuation payoffs following this history.
Then, if V 0 > V , players could leave Ē 0 and agree to move to E 0 instead. State consistency would
generalize internal consistency by allowing comparisons in such dynamic games.
One may envision many such games. For example, the stage game could entail players taking actions
that determine how they split a pie, whose size evolves dynamically, either as an exogenous Markov
process, or a controlled process that is affected by players’ actions in the last period. The statevariable Xt would then specify the size of the pie at time t, and one could imagine scenarios in which
any continuation equilibrium starting from state x could be scaled into a different continuation
equilibrium starting from another pie size x0 6= x.
This extension of state consistency to dynamic games, possibly including privately observed states
is a major direction to develop the ideas put forth in this paper.
Effort to Improve Real Cash Flow: The agent could affect the real cash flow by puttingsome privately observed effort. This extension is realistic in many settings (e.g., to study labor
incentives and taxation) and connects the model with numerous paper on dynamic moral hazard.
An earlier version of this paper (Strulovici (2011)) studies this extension: the agent produces
privately observed effort to affect his cash flow: dXt = (ξ − λXt + At )dt + σdBt , where A = {At }t≥0
is the agent’s effort process, adapted to FX , and the agent incurs an additively separable cost φ(At )at time t from effort At . Although they are more complicated, all the arguments and results ofthis paper continue to go through:41 assuming that φ takes the exponential form φ(a) = φ̄ exp(χa)
for some positive parameters φ̄, χ, the definition of state consistency and the form of renegotiationproof contacts (consumption and sensitivity) are unchanged and the agent’s cost of effort is now
41

The only significant change concerns the convexification argument. Convexifying the cost of effort across twocontracts leads to different paths for the cash flow process and affects consumption under each path. This issue
can be addressed by backing out the underlying Brownian path from the agent’s report process and expressing thecontracts in these terms.

40

proportional to the negative of the agent’s utility: φ(At ) = φ1 (−Wt ) where φ1 = s̄/χ > 0 and the
incentive compatibility condition (22) is replaced by the condition: u1 (s̄) = s̄(χλ + s̄)/(χ(θ − s̄).
The comparative statics also go through (some are harder to prove—see Strulovici (2011)) and newcomparative statics with respect to the cost of effort obtain.
Reporting Constraints Truthful renegotiation-proof contracts clearly remain truthful if the agent
faces additional constraints on cash flow reports and transfers. For example, the agent could beunable from over-reporting his cash flows beyond some upper bound. Such constraints reduce theset of possible deviations and facilitate truth-telling.
Self-Insurance and Hidden Savings Bloedel, Krishna, and Strulovici (2020) consider an alternative implementation of the model, in which the principal allows the agent to self insure by
investing in some asset whose return is predetermined by the principal. They show that equilibriaof the self-insurance problem are equivalent to the renegotiation-proof equilibria studied here. They
also study the possibility of hidden savings and show that this addition pins down the sensitivity
parameter of the contract and yields the contract studied by Williams (2011).

Appendix A: Proofs for Section 2
Proof of Proposition 3
Fix any x ∈ R and consider any w0 , w positive such that (x, w0 ) and (x, w) can both be reached bysome histories. Take some history (X0 , . . . , Xt ) leading to state (Xt = x, Wt = w0 ). Also let, for
P −1
As Zs denote the (random) output increment between time t and t + τ . Theτ ≥ 1, ∆τ = t+τs=t
continuation payoff of the principal following that history has the form
Π(x, w0 ) =x+ ᾱ2 − ᾱ1
1−ρPP
where ᾱ2 = Et [ τ ≥1 ρτ ∆τ ] and ᾱ1 = Et [ τ ≥0 ρτ Cτ ] and Et [·] is the expectation conditional on
history (X0 , . . . , Xt ).
Let (Ĉ, Â) denote the continuation of (C, A) following history (X0 , . . . , Xt ). Letting β = w/w0 , thetransformation (φβ (Ĉ), φβ (Â)) constructed in Section 2 is a contractual equilibrium starting from(x, w). Compared to (Ĉ, Â) it has the effect of scaling the effort process {Âτ = At+τ }τ ≥0 uniformly
by the factor β 1/γ2 and the consumption process {Ĉτ = Ct+τ }τ ≥0 uniformly by the factor β 1/γ1 ,and thus give the principal an expected payoff ofx+ β 1/γ2 ᾱ2 − β 1/γ1 ᾱ1 .
1−ρ

41

x
Consistency across promised utilities then implies that Π(x, βw0 ) ≥ 1−ρ
+ β 1/γ2 ᾱ2 − β 1/γ1 ᾱ1 . The

reverse transformation implies the reverse inequality. Hence,Π(x, w) =x+ α2 w1/γ2 − α1 w1/γ1
1−ρ

(36)

where
1/γ1

1/γ2

α1 = ᾱ1 /w0

α2 = ᾱ2 /w0.

Equation (36) shows that the conclusion of Proposition 3 holds for all w such that (x, w) can bereached by some history. For w such that (x, w) is not reached by any history, we can complete
the function Π(x, w) using the same formula. Moreover, notice that by consistency across outputs,
Proposition 2 implies that the coefficients α1 and α2 are independent of x, which shows that theconclusion of Proposition 3 holds for all histories.



Proof of Proposition 4
In any equilibrium, the agent’s promised utility w = W0 at time 0 satisfies:
w = u(C0 ) − h(A0 ) + ρE[W1 ].

(37)

We construct contractual equilibria of the following form:
1/γ2

(38)

1/γ1

(39)

At = a1/γ2 Wt
Ct = c1/γ1 Wt

where a and c are positive constants. In such equilibria, (38) and (39) imply that u(C0 ) = cw andh(A0 ) = aw. Combining this with (37) yields
E[W1 ] =

1+a−c
w.
ρ

To analyze incentive compatibility of the agent’s strategy, we must specify how the agent’s continuation utility evolves as a function of the observed increment D0 = A0 Z0 . Since A0 is proportional
to w1/γ2 and E[W1 ] is proportional to w, we conjecture that
W1 = w1−1/γ2 dD0(40)

for some positive constant d. Under this conjecture, we haveE[W1 ] = wdza1/γ2
where z = E[Z0 ] is a primitive of the model. This yields the relation
dza1/γ2 =

1+a−c
.
ρ

42

(41)

Now consider the agent’s incentive to put effort at time 0: for any effort level A, the cost of effort
is Aγ2 and the expected benefit in terms of promised utility is, from (40) discounted to time 0,ρw1−1/γ2 Azd. Therefore, the optimal effort level A0 solves
max ρw1−1/γ2 Azd − Aγ2 .
ASince γ2 > 1, the objective is strictly concave in A and the maximizer A0 is determined by the
first-order condition
ρzdw1−1/γ2 = γ2 Aγ02 −1 ,which yields

A0 =

zdρ
γ2



1
γ2 −1

w1/γ2 .

(42)

Comparing (38) and (42) yields
γ2γ2 aγ2 −1 = (zdρ)γ2 .

(43)

Rearranging (41) we havezdρ = (1 + a − c)a−1/γ2or(zdρ)γ2 = (1 + a − c)γ2 a−1 .

(44)Comparing (43) and (44) then yields(1 + a − c)γ2 a−1 = aγ2 −1 γ2γ2 ,which simplifies toa(γ2 − 1) = 1 − c.

(45)This shows that there exists a one-dimensional family of state-consistent contractual equilibria,
1−1/γ2

characterized by (38), (39), the dynamic equation Wt+1 = Wt

Dt d, and the relation (45)

between the parameters.
Equation (45) implies that c ≤ 1. This is expected, because cw is the consumption utility given
to the agent in period 0. If cw exceeded the promised utility w, the agent could just shirk andreceived in one period more than what is promised over his entire lifetime.
Also notice that for any give consumption parameter c, the agent’s effort parameter a is decreasingin γ2 , which is consistent with the fact that γ2 determines the agent’s marginal cost of effort.
Finally, fixing γ2 , notice that a and c are inversely related. This may be interpreted as follows:(1 − c)w is the amount of promised utility that is given to the agent in the form of expected
continuation utility as opposed to immediate consumption utility cw. The higher this amount, theeasier it is to incentivize the agent to work.
43

Appendix B: Proofs for Sections 4–7
Proof of Proposition 5
State consistency implies that, after any history leading up to some state (ṽ, x̃) = g ◦ (v, x) andcontinuation equilibrium (C̃, L̃), we haveΠ(C̃, L̃) ≥ Π(Φg (C, L)).
Suppose by contradiction that this inequality is strict. Monotonicity of Φ applied to g −1 impliesthatΠ(Φg−1 (C̃, L̃)) > Π(Φg−1 (Φg (C, L))) = Π(Φg−1 g (C, L)) = Π(C, L),
which contradicts the state consistency of C.



Proof of Proposition 8
We prove that if a strategy L gives a higher expected utility to the agent than another strategy
L0 under the initial contract, then it also does so under the new contract Ĉ. Suppose that L gives
higher expected utility than L0 under C. This means that
Z ∞
e

E

−rt

0

 
 
Z t
Ls ds
dt
u C(Ys : s ≤ t) −
0
Z ∞
 
 
Z t
−rt
0
0
≥E
e
u C(Ys : s ≤ t) −Ls ds
dt (46)00where Y and Y 0 satisfy the dynamic equations




Z t
dYt = Lt + ξ − λ Yt −Ls ds
dt + σdBt0

anddYt0 =



L0t +




Z t
0
0
ξ − λ Yt −Ls ds
dt + σdBt
0and the initial conditions Y0 = Y00 = x.
Now consider another initial condition x̂. The reporting processes under strategies L and L0 are
respectively



Z t
dŶt = Lt + ξ − λ Ŷt −Ls ds
dt + σdBt


0

and





Z t
dŶt0 = L0t + ξ − λ Ŷt0 −L0s dsdt + σdBt
0

subject to the initial condition Ŷ0 = Ŷ00 = x̂.

44

By construction, the virtual cash flow processes follow the equations
dỸt = dŶt − (ξ − λŶt )dt + (ξ − λỸt )dt
dỸt0 = dŶt0 − (ξ − λŶt0 )dt + (ξ − λỸt0 )dt
subject to Ỹ0 = Ỹ00 = x. Combining the last four equations pairwise yields




Z t
dỸt = Lt + ξ − λ Ỹt −Ls ds
dt + σdBt
0





Z t
0
0
0
0
dỸt = Lt + ξ − λ Ỹt −Ls ds
dt + σdBt
0

subject to the conditions Ỹ0 = Ỹ00 = x. By definition of the contract Ĉ, the agent’s expected utilities
using strategies L and L0 when facing contract Ĉ are identical to the expected utilities appearingin (46) and we have just shown that Ỹ and Ỹ 0 are subject to the same dynamic equations andinitial condition as Y and Y 0 . This prove that strategy L also dominates L0 under contract Ĉ. 
Proof of Proposition 12
Consider any reporting strategy L and let V λ (L) denote the agent’s expected utility when using
strategy L and facing contract C λ . Since the agent consumes Ctλ − Gt at time t and the agent has
utility u(c) = − exp(−θc), we haveλ

Z ∞V (L) = E

−rt

e



L
Utλ eθGt dt|L0

where Utλ = u(Ctλ ) and GLt =Rt

0 Ls ds is the gap. By construction of C

λ , we haveUtλ = λUt1 + (1 − λ)Ut2
for all t and report histories {Ys : s ≤ t}, where Uti = u(Cti ) for i ∈ {1, 2}. This implies thatV λ (L) = λV 1 (L) + (1 − λ)V 2 (L)(47)

where V i (L) is the agent’s expected utility with strategy L under contract C i . By assumption, thecontracts C 1 and C 2 are truthful, so for i ∈ {1, 2}, we have V i (L) ≤ v for all L and V i (L ≡ 0) = v.
Combining these observations with (47) yields V λ (L) ≤ v, and the inequality is tight if L ≡ 0. 
Proof of Lemma 1
Since C is truthful and state consistent, Ĉ is also truthful and we have Π(C) = Π(Ĉ) = Π(w, y)
by Proposition 11. Fix any λ ∈ (0, 1). From Proposition 12, the convexification C λ of C andĈ is also truthful and gives the same promised utility w to the agent given y. Moreover, strict
concavity of the utility function u together with the definition of C λ (Equation (17)) implies that45

Ctλ ≤ λCt +(1−λ)Ĉt ) for all t with a strict inequality whenever Ct 6= Ĉt . This implies that Π(C λ ) <λΠ(C 1 ) + (1 − λ)Π(C 2 ) = Π(w, y) and violates renegotiation-proofness of C (Equation (18)) unless
kC − ĈkL = 0.



Proof of Proposition 14
The principal’s objective is to maximize Π(w, x) with respect to u1 . From (14), and neglecting for
now terms that are not affected by the contract choice, this is equivalent to solving


Z ∞
αt
−rt
e
max
− E[Ct ] dt
u1
r+λ
0
where
αt =

E[log(−Wt )]χ

E[Ct ] = c1 −andE[log(−Wt )].
θ

Letting Zt = log(−Wt ), Itô’s formula implies that1dZt = (r + u1 )dt − σ 2 s̄2 dt − s̄σdBt .
2
Therefore,1
E log(−Wt ) = log(−w) + (r + u1 )t − σ 2 s̄2 t.
2
The principal’s objective is to maximize
−

c1 1
+
r
θ

Z ∞

e−rt E log(−Wt )dt.

0

Replacing c1 and E log(−Wt ) by their formulas in terms of u1 , t, and the parameters of the model,
and integrating the last term proves the proposition.

Proof of Proposition 17
The optimal value of s̄ is unchanged if we multiply (23) by r and get rid of the term r in the factor(r + u1 − 1/2σ 2 s̄2 ). The resulting objective function is equal to


1 2 2
r log(−u1 (s̄)) + u1 (s̄) − σ s̄ .
2
The function u1 (−s̄) is increasing in s̄. Therefore, the objective is supermodular in (s̄, r) and theresult follows (see, e.g., Topkis (1978)).



Appendix C: Proof of Theorem 2
To prove strong uniqueness, we can analyze the problem under any probability measure that is
absolutely continuous with respect to P. Consider the probability measure QY under which σ1 Y
46

is the standard Brownian motion. This measure is absolutely continuous because Y has constantquadratic variation equal to σ, and the drift of Y satisfies the standard integrability conditions.42
Equation (26) is an SDE with constant volatility coefficient and functional drift −L(t, ·).
Part 1. The boundedness assumed in the premise of Theorem 2 implies that condition (4.169) in
Liptser and Shiryaev (2001) is satisfied. Theorem 4.13 of Liptser and Shiryaev (2001) then implies
that (26) has a unique (in law) weak solution. This, together with Theorem 3.2 in Cherny (2002)
implies pathwise uniqueness, which implies that X is adapted to Y and the existence of a uniquestrong solution, by the Yamada-Watanabe principle.43
Part 2. Equation (26) satisfies assumptions (a1)–(a4) of Huang (2018): Assumption (a1) and (a2)trivially hold because volatility is constant, assumption (a4) holds because the term b in Huang’s
paper is identically equal to 0, and assumption (a3) is explicitly imposed in the premise of Theorem 2
and the premise of Part 2. Theorem 1.1 of Huang (2018) then implies pathwise uniqueness—which
implies that X is adapted to Y —and the existence of a unique strong solution.44
Part 3. Equation (26) satisfies conditions (C1)–(C3) of Bachmann (2020): Condition (C1) comesfrom the boundedness assumed in the premise of Theorem 2. Condition (C2) holds trivially because
the volatility coefficient is constant and positive, and Condition (C3) is explicitly imposed in thepremise of Part 2. Bachmann’s Theorem 1.5 then implies that pathwise uniqueness holds. Thisimplies that X is adapted to Y and the existence of a strong solution.

Appendix D: Proofs for Section 9 – Necessary Conditions for Incentive Compatibility
Lemma 2 For any truthful, renegotiation-proof contract, we haveZ ∞v̄y (w, x, x) = −v̄x (w, x, x) = −e−(r+λ)t u0 (Ct + Xt − Yt )dt.

(48)

0

Proof. The value function of the agent satisfies the optimization problem
Z ∞

−rt
v̄(w, y, x) = sup Ee (u(Ct (Ys : s ≤ t) + Xt − Yt )) dt ,L
42

0

The drift of Y depends on two components: the drift of X, which is linear in X, and the lying process L whichis assumed to be locally bounded.
43
This principle was stated and proved by Yamada and Watanabe (1971) for standard (as opposed to functional)
SDEs. However, their proof extends without any change to functional SDEs, a fact that is well-known and widelyused in the literature on functional SDEs. See, e.g., Cherny (2002, Figure 1) and Huang (2018, Theorem 2.8). I am
grateful to Nikolai Krylov and Stefan Bachmann for pointing this out.
44
See Theorem 2.8 in Huang (2018).

47

where Ct (·) is, for each t, a functional that determines the consumption provided to the agent attime t given past reports {Ys : s ≤ t}. If the initial cash flow is increased by ε, this affects the
distribution of future cash flows and, keeping the lying process fixed, of future reports. However,by a change of variable, one can control the path of the report process Yt , and make it independent
from the initial cash flow change. Recall thatdYt = [(ξ − λXt ) + Lt ]dt + σdBt .
Making the change of variable L̄t = Lt + (ξ − λXt ) − (ξ − λYt ), we get
dYt = [(ξ − λYt ) + L̄t ]dt + σdBt .

(49)

The agent’s strategy can be restated as choosing L̄, rather than L:

Z ∞
−rt
e (u(Ct (Ys : s ≤ t) + Xt − Yt )) dt .
v̄(w, y, x) = sup E
0L̄

subject to Y0 = y, X0 = x, (49), anddXt = (ξ − λXt )dt + σdBt .
dWt = (rWt − u(Ct (Ys : s ≤ t)))dt + St (dYt − (ξ − λYt )dt) .
If the contract is incentive compatible, it is optimal to set L̄t = 0 whenever initial conditions are
such that y = x. Section 9 established that the value function is differentiable with respect to x andy whenever x = y. An application of the Envelope Theorem (Milgrom and Segal, 2002, Theorem
1) then implies that v̄x (w, x, x) can be computed by evaluating the objective function at L̄t ≡ 0 or,
equivalently, under the report process Yt starting from y0 = x0 . Under this approach, W , Y , andC are independent of the initial condition x, andZ ∞dv̄x (w, x, x) =e−rt E [u(Xt − Yt + Ct (Ys : s ≤ t))] dt.
dx
0
Since the distribution of {Ys }s≤t is independent from the initial condition x, the inner derivative isequal todXt
0E u (Xt − Yt + Ct (Ys : s ≤ t)).
dx
The process X defined by the dynamic equation (6) is a generalization of Ornstein-Uhlenbeck
processes, which can be explicitly integrated:
Z t
Z t
Xt = e−λt x +eλ(s−t) ξds +eλ(s−t) σdBs
0

(50)

0

−λt and yields the formula of Lemma 2.
tThis implies that dX
dx = e48

Lemma 3 The following equality holds for any truthful renegotiation-proof contract:Z ∞
(−u1 )θ
−(r+λ)t 0
Eeu (Ct )dt = (−w).
λ − u10
Proof. Since u0 (Ct ) = −θu(Ct ),
Z ∞

Z ∞

Z ∞
−(r+λ)t 0−(r+λ)te−(r+λ)t E[Wt ]dt.
Eeu (Ct )dt = Eeθu(Ct )dt = θu1000Moreover,dWt = [rWt − u(Ct )]dt + St σdBt .
With the exponential utility specification, we have u(Ct ) = (−Wt )u1 . Letting ϑ(t) = E[Wt ] (ϑ(0) =w), this implies thatdϑ(t) = (r + u1 )ϑ(t),dt
and hence that
E[Wt ] = e(r+u1 )t w.
Integrating this expression over time yields the result.

(51)


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